's World of Ideas: Math, Science and technicalitiesAlternate Hexadecimal codes
Definitions of "Fortune/Luck" and "Money"
Division between man's soul and spirit: the 12 primary emotions
Metric Time
When did the new century/millenium begin?
Color mixing and perception
Math Trivia
•"Great Zero" (Lowest common multiple of all numbers from 1 to 100)
•Alternate negative number system
•Why is the "powers of ten" line asymmetrical?
•Grownup name of "googol"
•How many sides does a circle have?
Space & TIme
•Working definition of space & time
•What really is Time Travel?
•Multidimensional Time
•Bending of time: why faster than light travel would compromise causality
•P Theory/F Theory (Patrix/Father): The Third Continuum
•Representations of space and time in two scifi stories
Forgotten 4D object: the Duo-cone
Alternate Hex Codes
I never liked how the standard hexadecimal codes use A-F to represent 10-15. Because of the concept of alphabetic integers (link above), those to me represent 1-6. So I would have picked up the alphabet with the tenth letter. This would make it easier to figure out the decimal value, since I would think "10-15" when reading the letters:
A=J; B=K; C=L; D=M; E=N
Problem is, the next and 15th letter is "O" which looks like a "0" (zero) already used as an integer, of course. So I looked into other schemes. You can skip it and go to "P", substitute another letter like "X". or perhaps keep it as "F" (which in RGB color codes I tend to read as "full' as in "fully on" as representing 255 and the highest value. However, F0 and especially 0F fo not represent that). But I would like to keep the letters in sequence (like is for some reason the base would someday be increased, for instance).
I also looked into using other characters. Many of the ones on the keyboard already have other uses in computer code, so they wouldn't be good. Many others aren't on the keyboard, but then they would be complicated to type. (Alt codes). I thought of the Euro (€), which is being added to many keyboards, but it's too much like a "E". If it looked more like an "F", I would compromise. One idea is a ready-made "15" in a bullet or circle as a single character ⑮: U+246E; ⒂: U+2482. But then for that, we might as well use them for 10-14 as well. That might be a better idea, except that those characters are compact and do not look as different as different letters. I would think the best idea would be a simple 7/8 block (can't use full block because they would attach to one another creating a bar): ▉ (U+2589). So then, the smallest value would be "0", and the highest would be "▉" evoking binary! (0 and 1).
Some HTML codes from this page would look like: body bgcolor="#600020" text="#▉▉▉▉▉▉" link="#0080▉▉" vlink="#8020L0" (default vlink on other pages="#551j8k") alink="#▉▉0000" orange: FONT COLOR="#▉▉J000 light gray: "#JJJJJJ" light blue:
"#80L0L0" background color of my christian Index: "#n0m0m0"
(Another problem I have realized is that hex values are often written in lowercase, and the "l" (L) looks like 1).
Definitions of "Fortune/Luck" and "Money"
On one of my pages, I mention the concept of "fortune" or "luck" (which many Christians would oppose, insisting God is directly "controlling" everything in the world). I then came up with the following definition:
"Fortune, or 'luck'; I would define as 'an unknown principle of a disposition of a situation to a particular outcome especially to benefit or to adversity that is out of control of the person involved'; (rather than any magical or mystical meaning commonly associated with the words). The emphasis is on 'unknown' and 'out of control of the person'. Whatever exactly, or directly causes things, even if it is God; no one can deny that much of the circumstances of life fit this description, from our perspective."
Later, in a debate with someone questioning my "introverted thinking" cognitive preference, the rhetorical question of "what is money" came up! I quickly responded:
What we call money is a "symbol or representation of wealth" used for the purpose of "trading". Notice, ever dollar bill is called a "NOTE". It is a reminder that it represents something, and you give it to a person in exchange for something else, a product or a service. Along with these "notes", you also have metal coins, and before those, material such as gold was used, and even before then, goods themselves were what were traded. It was easier to come up with these coins and notes to represent material value.
Division between man's soul and spirit
Ever wondered what was really the difference between man's "soul" and "spirit"? Sometimes, they seem synonymous, and there are battles between "dichotomists" and "trichotomists" over that. Both are said to be the "invisible" part of man, and how can we know where one ends, and the other begins to identify them clearly? A good guideline in differentiating between them I have found in the works of the late Christian psychologist Conrad A. Baars (Feeling and Healing Your Emotions Plainfield, NJ, Logos International, 1979). He divides our 12* basic emotions into "humane" emotions, (love/hate, desire/aversion, joy/sadness), which are ennobled by our or "intellect" ("intuitive", or "contemplative" mind); thus making up our "heart"; and also our "utilitarian" emotions (hope/despair, courage/fear, peace*/anger), which aid our "reason" ("working" or "discursive" mind) thus making up our "mind". ("intuitive" comes from a Latin word meaning "look" or "view", and "intellect" from "to read between", both as opposed to simply "reasoning")
"Upwardly" he says, "the humane emotions are intimately linked with our spirit, and the utilitarian emotions with our reason" [i.e. soul]. Downward, both groups are linked with our body. (p.33). The humane emotions are from our "pleasure appetite" and cause inner movement within the psyche. They are our responses to what we perceive as "good" or "bad". Our intuitive mind also receives its knowledge from such sources as nature, the arts, faith, and directly from God through the Spirit, thus echoing the biblical statement. The utilitarian emotions of our "utility appetite" move us to action to make life better or respond to threats to our happiness or well being. Thus, they are concerned with mundane things; what is useful or harmful. It's the humane emotions that distinguish us from animals (hence, "humane"). While they certainly share the utilitarian emotions (anger, courage, etc) with us, the other set of emotions are not "ennobled" in them, being that they have instinct to guide them. Since we have those emotions, our instincts are undeveloped or "sophisticated" (its character altered).
>So this gives us a good idea of how to distinguish our soul from our spirit: just think of the emotions associated with them!
I find it helpful to categorize the different pairs into categories:
| description | Anticipating (future) | Present reality | reactive (past) | |
| "humane" emotions | "intellect"["to read between"] ("intuitive", ["look" or "view"], or "contemplative" mind) "heart" | desire/aversion | love/hate | joy/sadness |
| "utilitarian" emotions | "reason" ("working" or "discursive" mind) "mind". | hope/despair | courage/fear | peace*/anger |
*Baars does not recognize an opposite of "anger", which he calls the "ultimate emotion". But it seems "peace" or "contentment" would fit. Anger is a "sense-evil" emotion sort of like an active, charged version of sadness, and a temporal cousin to hate. So its opposite would be similarly related to love and joy. "Peace", as it is defined in the Bible is a more spiritually charged form of joy, and is connected with love. It is needed when the other utility emotions are not able to remove the cause of pain or unhappiness, or when something gives you pleasure apart from the intuitive mind. The proof is that animals such as our pets would have the sense-evil reaction of anger if teased, but if petted, a sense-good reaction that is not the "humane" love or joy, and certainly not hope or courage. They are then peaceful. Baars and his colleagues considered this state (which they referred to as "meekness") as not an emotion, but as a spiritual state. But this would probably result from the fact of anger appearing to be the "ultimate emotion". It's opposite then, may appear not to be an emotion at all. But its presence in animals proves it must not be "spiritual". The "peace that surpasses understanding" given supernaturally to humans by God would be the spiritual state.
Metric Time:
This I had always thought if all of the other units of measurement were supposed to be converted to metric (based on divisions of 10), then why was time omitted? If we got rid of the "inch" for space measurment, and had a "centimeter" instead; and other units like grams, liters, and Celsius (centigrade) degrees, then why stick around with the old "second"? In fact, in astronomical calculations, where hours, days, weeks, etc. are irrelevant, they have made a metric system based on the second (milliseconds, microseconds, etc). They didn't make "kiloinches", "decifeet", "megapounds", or "centigallons", but rather created whole new units, based on logical points of measurement, like the freezing or boiling point of water being 0 or 100 degrees Celsius. The same units were then projected back to absolute zero, and when begun on that scale, called Kelvins.
A type of metric time I first saw when working the Census was using ".5" instead of ":30" and ".25" instead of ":15", etc. Nice idea, but I wish it was taken further. So I figured, why not take the most obvious and necessary unit of time on earth, the day, and divide it down into new units. (Units larger than a day wouldn't work well, because of the seasons and year). I started by dividing the day by 10, and continuing to divide each unit by 10, until I got to a unit of time most comparable to the second. here, in minutes:seconds--
1/10=144:00
1/100=14:24
1/1000=1:26.4
1/10000=00:08 (8.64)
1/100,000=00:0.864
So the new metric second would be 1/100,000th of a day, and slightly smaller than a second. Sort of like the temporal counterpart to the centimeter. Since below "centi" (10-2), only powers divisible by 3 were given Greek prefixes (-3, "milli"; -6, "micro", etc), there was no way to name this new unit with a standard SI prefix like the others. I suggested a hybrid prefix, centimilli, meaning of course, one hundredth of a thousandth. So it would be "centimilliday". Hour-like units would be 1/20= 00:72:00, ("semideciday"), at 72 minutes. Or 1/25= 00:57:36, which is about 2½ minutes short of an hour.
Here is the conversion between our time (military and 12 hour) and the new time.
| 0000 (12:00A) 00:000 00:14:24 1000 12:15 ≈ 01:041 00:28:48 2000 12:30 ≈ 02:083 00:43:12 3000 12:4503:125 (1/32) 00:57:36 4000 (1/25) 12:59:24 04:125 1:00 ≈ 04:166 1:12:00 5000 (1/20) 1:26:24 6000 1:30 06:250 (1/16) 1:40:48 7000 1:55:12 8000 (2/25) | 2:00 ≈ 08:333 2:00:36 08:375 2:09:36 9000 2:15 09:375 2:24:00 10000 2:52:48 12000 (3/25) 3:00 12:500 3:36:00 15000 3:45 15:625 3:50:24 16000 (4/25) 3:59:2416:625 4:00 ≈ 16:666 4:48:00 20000(5/25) 5:00 ≈ 20:833 5:45:36 24000(6/25) | 6:00 25:000 6:43:12 28000 (7/25) 7:00 ≈ 29:166 7:12 30000 7:40:48 32000 (8/25) 8:00 ≈ 33:333 8:24 35000 8:38:24 36000 (9/25) 9:00 37:500 9:36 40000 (10/25) 10:00 ≈ 41:666 10:33:12 44000 (11/25) 10:48 45000 11:00 ≈ 45:833 11:31:12 48000 (12/25) | 12:00 50:000 12:28:48 52000 (13/25) 1300 (1:00P)≈ 54:166 1312 55000 1:36:24 56000 (14/25) 1400 (2:00P)≈ 58:333 1424 60000 (15/25) 1500 (3:00P)62:500 15:21:36 64000 (16/25) 1536 65000 1600 (4:00P)≈66:666 16:19:12 68000 (17/25) 1648 70000 1700 (5:00P)≈70:833 17:16:48 72000(18/25) | 1800 (6:00P) 75:000 18:14:24 76000 (19/25) 1900 (7:00P)≈79:166 1912 80000 (20/25) 2000 (8:00P)≈83:333 20:09:36 84000 (21/25) 2024 85000 2100 (9:00P) 87:500 21:06:24 88000 (22/25) 2136 90000 2200 (10:00P)≈ 91:666 22:04:48 92000 (23/25) 2248 95000 2300 (11:00)≈ 95:833 23:02:24 96000 (24/25) |
Come to find out, that I am not the first one to think of this at all! This system was actually planned back in France around the same time as the rest of the metric units (1700's) and even implemented before the others, but then decided against. I guess because of the almost universal worldwide use of the standard 12-based system. The system is now called "Decimal Time", and I even found a website and forum on this, (http://www.decimaltime.hynes.net) where I shared my ideas. One of the discussions was a name for the new metric second. Suggestions have been "tick" and "beat" (the heartbeat is said to be the same length as this 100,000th of a day!) I even hear of, and went to see a watch by Swatch using this system, in its Times Square store! Also, astronomers, beginning with John Herschel (Outlines of Astronomy) in 1849, use another version of this called fractional days to represent time in their writings. (http://decimaltime.hynes.net/fraction.html).
Also, the decimal time they use is apart of what they call "Universal Time" (UT), and begins at 8PM at a particular time zone, rather then midnight in every time zone.
When did the new century/millenium begin?
Now several years too late is an explantion I thought of, of the question of when the millenium and new century began.
The distinction is one of cardinal vs ordinal. Cardinally, 2000 began the set of "2000's" (with the counting numbers of zero, one, two, etc); and hence was the first year of the new millenium. But ordinally (which begins with "first", not zero, and then continues "second", "third", etc), it closed out the 20th century) which encompassed most of the 1900's series years). The problem is that years themselves are cardinal, but centuries are ordinal. So the "hundreds" series do not lne up with the century numbers, except for the one year which is the multiple of 100, which starts the cardinal series, and ends the ordinal century.
Monochrome yellow vs. filtered or red+green mixing
Following the development of LED technology (from red, yellow and green only, to the addition of blue, making possible all the rest of the colors), I learned about "additive mixing" versus "subtractive" mixing. The standard color-wheel based system, by which paints and inks are mixed, and the primary colors are red, blue and yellow, differs from the way light is mixed, where the primaries are red, green and blue. I always wondered why the primaries of the color TV screen (usually printed on the set as part of the TV brand's logo) had green, instead of yellow. Green is yellow mixed with blue, isn't it? But in reality, even the red and blue of the TV screen are slightly different than the red and blue you see on those ink bars next to the pictures on some of the color pages of the newspaper (such as the Sunday funnies, or ad circulars). The red appears slightly yellowed, and the blue looks "electric", like it has a tinge of red. These however are the primary colors of light, at least to our eyes. What you think is a primary, non-yellowish red ink actually is reflecting a small amount of blue light along with the red. (reflecting, hence "subtractive", because color is made by filtering out what is not reflected). This is called "magenta" or "fuschsia", though the primary ink color is deeper than the color with the hexadecimal RRGGBB value of FF00FF of the color screen. It is really more like FF0080. Likewise, the "pure" looking blue is reflecting a small amout of green. This is called cyan, and likewise differs from the 00FFFF of the screen. It is really a color called azure, and is 0080FF. So yellow is an equal mix of red and green: FFFF00. That's why yellow light reflects so well off of red and green surfaces. Here are examples I am talking about: Subtractive: █ █ █; Additive: █ █ █
If you look closely at these bars, you can see two primary colors of each pixel lit for the subtractive colors which mix to form the overal color when you do not look at it closely; while the additive ones have only one pixel lit, representing the color you see upfront. The white background has all three colors lit. So if you're mixing light, such as LED's or TV pixels, the primaries are on the right. But to get those colors on a printed page, you mix the ones on the left; in which case you will see the colors on the right are made from small dots of different colors mixed together, or sometimes a transparent "stain", especially the yellow being printed over the blue to turn it green, or fuchsia to turn it to normal fire engine red.
But what about those brilliant amber light tubes that are seen under overpasses and other places? (A few places use them as street lights, such as Oyster Bay and Babylon townships in Long Island, and they have lined entire tunnels at times, such as the Brooklyn Battery Tunnel!) They distort ALL colors, including red and green, which become dull brownish looking colors. It looks so funny going through a tunnel bathed in that light, and you have on a bright red shirt, that now looks little different from any other color. You're basically in a black and yellow ("B/Y") world! (Much like the old Game Boy!)
Well, yellow can be made by mixing red and green, but it can also be a monochrome wavelength of light. In looking at LED's, you see this, because the color is measured by the nanometer. The bright 590nm LED's, that began appearing on message signs in the 90's, are the same color as the low-pressure sodiums, which are the awkward street light bulbs (The peach colored more common street lights are high-pressured sodium).
So I wondered how could that be, if yellow is actually not a prime color. I then realized, and thought up this nice rhyming illustration: vision is caused by
| EMISSION —> TRANSMISSION —> RECEPTION —> PERCEPTION |
Light is emitted by whichever source, then it travels through whatever medium (air, vacuum, glass, etc. in which it can be altered somehow), and then enters the eye, and then an image is sent through the nerves to the brain. The eyes consist of the red, green and blue "cones", which receive the waves of light. (This page http://casa.colorado.edu/~ajsh/colour/primary.html#diagram suggests that the "fundamental primary" green which lies way off of the normal spectrum of visibility, is closest to 497 nm on the spectrum, which is basically light blue!) Light of any wavelength will stimulate certain cones more than others. So an LED that makes yellow by mixing red (600's nm) and green (lower 500's) will send both wavelengths to your eye, and stimulate both the red and green cones, and you'll perceive "yellow". Also, if you take white light, and place a "yellow" lens over it, it will filter out the blue, and allow the red and green to pass, and reach your eye, and you'll again see yellow. The monochrome amber of the 590nm LED or low pressure sodium will consist of one wavelength, however, it will still stimulate both red and green cones. That is why you perceive it as yellow like the other lights. However, it will look much more "saturated" than the other yellows, and not reflect off of as many things.
This definition of perception also answers the old question: "If a [noise] goes off somewhere where no one hears it, does it really make a sound?"
Well, it would still vibrate the air molecules in waves radiating outward. Even if these waves never encounter any eardrums, they still did vibrate the airwaves, which is what "sound" is in one sense! We could insist on defining "sound" only as when it vibrates an eardum and is perceived by a living brain. So all a person has to do is pick from the two definitions, and they will have their answer.
"Great Zero"
69,720,375,229,712,477,164,533,808,935,312,303,556,800
(41 digits; read as "69 tridecillion, 720 dodecillion,... etc.)
This is the lowest common multiple of all numbers up to 100. I always wondered what the number would be like, since it has one property zero has in our everyday multiplication, and that's all numbers from one to one hundred go into it. The easiest way to get a number like that would of course to be to multiply all numbers from 1 to 100. This is called the factorial of 100. But I wanted the smallest number possible. So the way to do this was to multiply the highest powers of all prime numbers. So it's 64×81×25×49×11×13...and then all prime numbers up to 97. 100 will go into it, because the 64 contains 4, and then multiplying it by 25 gives you something divisible by 100. And so with so many other numbers. There are many numbers higher than 100 that go into it as well. Also, since all two digit numbers go into it, if you replace the two zeroes at the end with any two digit integer, that number will go into the new number as well.
It was in the 90's; I don't have a computer yet, but I got the idea and someone let me use the computer at their office. I don't even remember what program that was, she just brought up something and I could multiply all the numbers. The calculator in my Windows accessories began doing that "e +40" stuff where it leaves out digits and makes it a huge decimal, after 73. Whatever program the office computer had (I just remember it being done on a blank screen) didn't do that. I then printed it out.
Just for kicks; I multiplied the whole number by the next prime number, 101 sometimes later. That way, cool numbers like 1111 and its multiples would go into it. 111 already goes into it, because that's 3×37. 1001 goes into it because that number is 7×11×13; and by extension, multiples like 111,111, which is 111×1001. So the number with 101 multiplied in is:
7,041,757,898,200,960,193,617,914,702,466,542,659,236,800
(7 quadridecillion...)
Alternative Negative Number system
These numbers can also help get an idea of an alternate negative number system I had thought of. Instead of the number line to the left of zero being a mirror image of the positive numbers, it would be a backwards extension of the positive numbers. So 0 - 1, which is the same as ...0000000000000 - 1, would be "...999999999999", instead of "-1". subtract another one, and it would be "...9999999999998"., 0 - 10 would be ...9999999990. 0 - 100 would be ...9999999900. Think if you had a very large number like "googol", discussed below. That's a 1 followed by 100 zeroes. Subtract 1 from it, and you have a string of 100 nines. Subtract 10 and you have 99 nines followed by a zero. Subract 100, and you have 98 nines followed by two zeroes. And So on. The only problem, as big as a googol is, many numbers do not go into it. Its prime factors are nothing but 2 and 5. No other prime number (3, 7, 11, etc) or their multiples will go into it. But with these "great zero" numbers, all numbers from 1 to 97 or 101 will go into it, plus many more; so going backwards from it, you get a sense of this new "negative number" system I am talking about. So the next number divisible by 3 going backwards, instead of "-3", will be "....97". The next multiple of 7 will be "...93". It's as if I have looped the number line around so that zero is both the beginning and the end (infinity), to a limited extent. Hence in the new "negative" numbers, the infinite number of digits to the left of the ones place (still the rightmost place by default). I had always tried to imagine a hypothetical "end" of the number line. You get one last number consisting all of nines, and then add that final one, and get zeroes. Of course, both the nines and the zeroes would have been infinitely long, to the left. And this is what I have here, but with zero as the infinitely far end. The beginning and end are right before you; it is just a circle with an infinitely long circumference. This is another mathematical concept I mention below.
If you were to hold the zero in the ones place, and realizing that there are an infinite number of zeroes to the left; then subtract 1, but treating it as a very large multiple of a large power of 10, then instead of "-1", you keep putting 9's, and then subtracting 1 from the next zero to the left, since you had to take away from the previous zero. You will end up with an infinite series of nines to the left. And so on. One oddity, is that ...9999999999 appears to consist of all nines, and any such number is always divisible by both 3 and 9. (9×...1111111111). But this is not the case here, because ...99999999999 would be 0 - 1, and zero itself is 3×0 and 9×0. In fact, the ...9999999999 number would be prime! Every other number meets at ...00000000000, so only 1 would go into ...99999999999. It is really -1, remember! ...11111111111111 would be the number divided by 9, but that number would be infinitely far away. It would basically be the new -.1111111111..., which is -1/9.
Multiplication and division really don't work with these numbers. After all, they have infinitely many places to the left of the decimal point. You couldn't even start multiplying them. (where numbers like 1/9 or 1/3 have infinitely many places to the right of the decimal, you could start multiplying them, and realize that it just continues on left to right forever). Thus, it would not help define the square root of -1 (i.e. "imaginary" numbers) either. So this new negative number system is primarily for addition and subtraction. (The only way to cross over to the negative range with division or multiplication is by directly using negative numbers, and then it works exactly the same as dividing or multiplying positive numbers; only the - or + sign changes). Just thought it would be an interesting alternative. Would make subtraction into the below 0 range easier.
Why is the "powers of ten" line asymmetrical?
Another kind of number line was the powers of 10, with 1 in the zeroes place as 100, and then each positive power of 10 corresponding to the same number of zeroes after the "1". However, on the negative side, it seemed off. While "ten" (10¹) is "10.", "tenth" (10-1) is not ".01", but rather ".1". Likewise, "hundred" (10²) is "100."; while "hundredth" (-2) is not ".001", but ".01". "Thousand" (10³) is "1,000." and "thousandth"-3 is not ".000,1", but ".001")
Here is a larger sample:
| Million | 10 6 | 1,000,000 |
| hundred thousand | 10 5 | 100,000 |
| ten-thousand | 10 4 | 10,000 |
| thousand | 10³ | 1,000 |
| hundred | 10² | 100 |
| ten | 10¹ | 10 |
| one | 100 | 1 |
| tenth | 10-1 | .1 |
| hundredth | 10-2 | .01 |
| thousandth | 10-3 | .001 |
| ten thousandth | 10-4 | .000,1 |
| hundred thousandth | 10-5 | .000,01 |
| millionth | 10-6 | .000,001 |
| ten millionth | 10-7 | .000,000,1 |
This appears to be a "reflectional symmetry" that is broken.
I found it so interesting to see another kind of "mirror image" of the numbers, even down to the grouping of zeroes in threes by the commas. But it was so off! It made me wonder for a while, should we have started off with ".1" as "oneth", and then make "tenth" the next negative power? But that would throw a lot else off!
But the reason why it appears broken, is that we are looking a the axis of this symmetry, as the decimal point. But actually; it's —the ones place, itself! "One" is what is defined as 100, and zero is the center of the number line, or the axis or fulcrum in which the symmetry hinges. The reason it appears off, is because the decimal point is noted next to (to the right of) the ones place; but it is really supposed to be marking the ones place itself; not the space between the ones and tens place. If we had developed a notation system using an accent above or below the ones place, then it would look more symmetrical. Then, "01", with the "0" so noted as the ones place, would be "tenth", instead of ".1". So what mathematicians have often done, is to fill in the ones place with the "0" to the left of the decimal point, so you end up with a matching numer of zeroes even with the decimal point:
| tenth | 10-1 | 0.1 |
| hundredth | 10-2 | 0.01 |
| thousandth | 10-3 | 0.001 |
| ten thousandth | 10-4 | 0.000,1 |
| hundred thousandth | 10-5 | 0.000,01 |
| millionth | 10-6 | 0.000,001 |
True "grownup" name of "googol"
Also, other technical number information: the proper name of number they call "googol" (10100) is ten duotrigintillion. Others have pointed this out (including Wikipedia, for starters), but I figured it myself before the Internet age by realizing that after the one, it will consist of 33 groups of three zeroes, with one left over; so it will be "ten-something-illion". Then, notice the powers of 10 divisible by 3 have Latin Prefixes attached to the "illion": billion, trillion, quadrllion, etc. The Latin number goes up with each set of three zeroes added. But it is offset. Billion is not two sets of zeroes, (106) but rather three (109), trillion is four sets (1012), etc. This is because "million" started out as 106, but the sets of zeroes were six. So billion was two sets of 6 zeroes, or 1012, and so on. This was the old British system, but the American system started with the same "million" and began increasing by sets of three. So 33 sets of zeroes would have the Latin term for "32", rather than 33, and this is duotrigint.
How many sides does a circle have?
When I was in the later grades of Grammar school; I had become interested in polygons. During that period, I had remembered an old Sesame Street segment where Ernie asks Bert how many sides a circle had: One? None? Well, trying to draw polygons, and realizing that the more sides you add, the more it looks like a circle; I figured it must really be ∞ sides! In technical terms, a circle can be viewed as the "limiting case" of a regular polygon with a fixed radius. As the number of sides approaches infinity, the length of the sides decreases towards zero. The angles approach 180°, and the figure's sides smooth out to a perfect circle! Notice, they approach the values of 0 or 180. They don't reach it, since you can't actually reach infinity (as if it were a single point anyway). So these values are called asymptotes. On the other hand, if we now fix the length of sides, the radius now approaches infinity instead, and the adjacent sides flatten out to an infinite straight line tiled with the original lengthed line segments (the "Sides"). This has been called an apeirogon. (It is the shape of the new "number line" I propose above with zero as the common starting and ending point, but the line still extending infinitely both ways!) Just like "pentagon" means "five sides", and hexagon" means "six sides" ("-gon" actually refers to the number of angles, as it will always equal the number of sides in 2D geometry. "gon" is actually directly akin to the word "knee"); "apeiro-" is the prefix for infinity. It basically means "without perimeter".
Since this term is used only for the straight line case where the length of sides was fixed; I tried to come up with other terms in a math discussion for the fixed radius "circle" version, such as "closed apeirogon", or "achanegon"; "achanes" being another Greek term for infinity (http://www.kypros.org/cgi-bin/lexicon); this one meaning "roofless" (according to this site http://www.quantavolution.org/vol_13/firenotblown_16.htm; this was the ancient terms used by Sophocles, Fragment 1030, and Kerenyi, The Gods of the Greeks, p. 270 regarding the dancing floor at The labyrinth at Knosos).
We end up with a paradox; because the circle, and the endless straight line are supposed to be the same object! What has happened, is that the straight line, in which the lengths of the sides remained fixed is an infinitessimal projection of the perimeter of the circle! The sides in the "circle" projection had shrunk down to zero in length, remember! This makes sense, as every line segment is considered to be composed of an infinite number of points, which are zero in length. It has to be, as mathematically, there is no limit to division of length. Take any object with a width, or area, those lengths can always be halved, tenthed, hundredth, etc. Take any space between objects, and it can always be halved, etc.* A polygon is defined by how many of these straight lines meet at angles. The circle's perimeter does not have straight lines meeting at angles, but is rather a smooth curve, still defined as a set of points. So all you have to do is consider the circle's "points" as its "sides". If you blow one of these sides up by a magnitude of infinity, you will see it in a series of line segments at 180° with an infinite radius! An infinite number of these line segments still occupies a single point on the circle! In connection with this, you may wonder how a straight line composed of line segments at 180° can stretch out to a curved, closed figure. Well, just as I just said, that infinite line is really only one single point on the circle. Since the radius is fixed in the circle, then at any given angle, the same radius length from the center, you will find a point, which will magnify to an infinite line. The entire set of these infinite points is the circle! While the lines segments are "next to each other" (adjacent) at 180°; the individual points on the circle are not "next to each other". If you take one point, of zero length, and place it "next to" another point, also of zero length, they won't sit beside one another; but occupy the same spot! (Unless you have some amount of space between them). It would take an infinite number of them to reach the "next" place; and even that is undefinable, as they do not fill any space for there to be any "next place". "Next to" nothing is still nothing! Only when magnified infinitely, do the points become line segments that can lie "next to" each other. It is the fixed radius that determines that at any angle; even thousandths or millionths of a degree, and smaller, there will be one of these "points" that magnify to an infinite line. In other words, a finite radius in an ∞-gon forces the object into a finite convex hull (surface closed around a point) whose perimeter is set by the radius. The finiteness of this perimeter forces the line segments and even the infinite line they make up, down to points, in order to "fit". These will form the entire set of points that same radius length from the center, and this is the definition of a circle! This paradox is the result of messing around with zeroes and infinities in the same equations!
We can see how it takes shape by looking at polygons with very large numbers of sides. Remembering that each side is bounded by congruent angles bisected by radii, which with the side form a triangle. The sum of all angles of a triangle must add up to 180°. The easiest we can start with is a hexagon. Each angle formed by adjacent radii meeting at the center will be 60°, and the other two angles will add up to 180-60, or 120. They of course will also be 60°, and the triangles will be equilateral, with both the radius and the side all being congruent. The angles the radii bisect— the interior angle of the polygon itself, will of course be 120°. So the radii, as the "spokes" of a wheel, will form isoceles triangles, which in this case is also an equilateral triangle, but will become thinner as S (the base) decreases, or r (the vertical sides of the triangle) increases. The angles at the center (C) will always be 360/n (n=number of sides), while the angles adjacent to the side (A) will be 180-C. Below, we can look at several polygons; fom the common everyday ones with small numbers, and increase n to ridiculously large numbers, for hypothetical polygons that would all be indistinguishable from a circle.
polygon name n (Number of sides) C (angle at center: 360/n) A (interior angle of polygon: (180-c) S (length of sides) if r=1
(2 cos A/2)r (radius) if s=1 (.5/cos A/2) common "shape" name trigon 3 120° 60° 1.732 0.57735 triangle tetragon 4 90° 90° 1.414 0.7071 square pentagon 5 72° 108° 1.17557 0.85065 hexagon 6 60° 120° 1 1 octagon 8 45° 135° 0.765 1.30656 decagon 10 36° 144° 0.618 1.618 hecatontagon 100 3.6° 176.4° 0.06282 15.918 hecatonogdocontagon 180 2° 178° 0.0349048 28.649 triacosiahexacontagon 360 1° 179° 0.017453 57.296 chiliagon 1000 .36° 179.64° .006283 159.155 megagon 1,000,000 .036° 179.999964° .00000628318 159,915.5077 gigagon 1,000,000,000 .000036° 179.999999964° .00000000000000628318 159,155,077.52 teragon 1,000,000,000,000 .000000036° 179.999999999964° .00000000000000000628318 159155077524.438 ["Great Zero"]-agon) 69720375229712
477164533808935
312303556800 .000000000000000000000
0000000000000000051634°179.99999999999
999999999999999
99999999999948365°.00000000000000
00000000000000
000000000000901110963070498650
62522983129408
834xxxxxxxxxxx.xapeirogon ∞ 0° 180° 0 ∞ circle
Everything with n as 1000 and above, with a radius that fits on a page, will look like perfect circles. n=100, 180, and 360 will look like circles with bumpy perimeters. 1000 and one million, the "sides" might be smaller than the ink dots making up the circle. The next two, the sides would be smaller than light waves. For the largest finite number; "great zero", the sides would be smaller than the strings that make up the fabric of space.* Magnify one of the sides to 1 inch (or centimeter, etc) , you will see what looks like an infinite straight line tiled with that side and its adjacent neighbors. Eventually, you will move along the perimeter in sizeable numbers of degrees, but the more sides, the longer you will have to travel to get to the next measurable "dot" or degree. A chiliagon whose sides were one inch long, would be have a radius of about 26½ feet, and be 53 feet in diameter! A megagon whose sides were one inch, would have a radius of 2½ miles, and be 5 miles wide. A gigagon with one inch sides would be just over 5000 miles wide (a little bigger than the planet Mars), and a teragon, 5 million miles. With 69 tridecillion sides, it would be almost 30 trillion light years wide; thousands of times larger than the known universe! A single degree would be wider than the universe. Starting from the center (also beyond the visible universe), in any angle you travel in, you will find what looks like an infinite series of line segments, (and keep in mind, one inch each!). If you tried to travel along this line, you would basically never be able to get to the point where the radius one degree; or even a thousandth of a degree, away intersects the line. It's just too far away for us to reach with the limitation of space and time. And this with a number that is still technically finite! The "[googol]gon" would even be bigger than that. (Or the sides smaller with a 1 inch radius). Similar figures for the teragon in the table, but with around 100 0's or 9's. One inch sides producing a radius and diameter trillions of trillions times larger than the universe, and one inch radii producing sides trillions of trillions times smaller than the smallest hypothetical particles of the universe. These are even 'more than perfect', perfect circles, since the smallest particles making up any circle we draw would not even be that small! The chiliagon for all of our practical usages, is a perfect circle. You probably would only barely be able to feel the angles or make out the ends of the one inch sides of the 53 foot model. They would all be so slight as to make a smooth curve. Higher than that, it will be indistinguishable from a circle in our perception.
So, infinity is the extreme case where our everyday logic breaks down, and you have an infinite series of infinite straight lines at every infinitessimal point. You would never be able to get to the next point traveling on one of these lines. You would never leave the radius at the angle you started from. They would not even be "too far". In this case, two points separated by any degree would really not even be connected to each other. There would be an infinite amount of space between them. In mathematical terms, the infinite line tiled with countable line segments is designated with an infinite quantity known as "aleph 0". However, when we "zoom out" to the whole circle, we are actually making a "quantum leap" from aleph 0 to another infinite quantity called simply "C" (for "continuum"). even though they are both infinite, C is actually "bigger" than aleph 0. While aleph 0's units are countable, C's "units" (the points), are not countable, because of the infinite density of points described above. There is no way to even begin assigning numbers to each point to "count' them. Hence, the paradox is the result of leaping between two types of infinity. One infinity we "approach" forever, by adding countable units, and the other, where we have presumed to have "reached" infinity; yet have made an infinite "leap" to do so.
The circle is made to be infinity, rather than simply assigning it as 1,000 or 1,000,000 sides, because it is only a hypothetical concept. One million sides is basically indistinguishable from one billion sides, or larger than that, or all of the numbers inbetween. The figure is already at its limiting shape, and it's only the scale it is constructed on, and the perception of those measuring it, that make them possibly different. So we might as well say that the perfect circle is the entire set of points a given radius from the center, which hypothetically is infinite, even if we can't measure anything that small or large.
Likewise, by extension, a sphere is an achanehedron (infinite number of points for its "faces"). A hypersphere is an achanechoron, and a hyperhypersphere is an achanetesseron. ("hedron" means "seat", for the "faces" of 3D closed figures. Other extended geometrical terms and their logical extensions you can find used online: "choron" means "room" —connected to "choreography", for the 3D "cells" of a 4D polytope; and -tesseron is for the 4D hypercells of a 5D polytope. "-tope" is the n-D collective suffix for facets in any dimension. "facet" is the collective term for the "sides" that bound closed figures in any dimension: lines segments in 2D; "faces" or plane segments in 3D; 3D "cells" in 4D, etc. ).
*If string theory is true, there may actually be a "smallest length" called the "Planck length" (10-33 cm). This is so, because the theory proposes the "fabric" of space as itself consisting of loops of string that are that length. Smaller that that, the concept of "space" breaks down, and there is no longer any medium to make any measurements on. Still, numerically, for the sake of this discussion, we can conceive of a tenth of that size; 10-34 cm.
A working definition of space and time
SPACE:
If I take a ruler and show one foot, or one inch, that is based on this wood, plastic or metal ruler, with markings etched into it. I can point to objects in the room several feet away, or a building a mile away in one direction, a body of water a mile away in another direction, or some other land feature in another direction. All of this is planted on raised levels of earth exposed above the sea level which remain the same shape. And these lands are all positioned on the globe of earth. They are basically stable and do not move, so this is how space (places) are reckoned on earth. It is all MATTER.
Now beyond earth it is not as definite. The entire surface of the earth is "moving"; rotating around the axis. Then, the earth itself is revolving around the sun, and the sun revolves around the center of the galaxy, and even the galaxies are moving in relation to one another. So as one book says you cannot drop anchor in the universe and determine a universal "location" here. So location and distance can be defined by local matter. Space is marked by matter in a given arrangement. So the definition of space we arrive at is the medium in which the relative positions of matter are measured. (As another book says, if you had a universe with no matter in it at all, distance and location would have no meaning or be undeterminable).
TIME:
I know one second or one minute has passed by looking at a clock; once again, matter. Naturally, the point on the surface of the earth I am standing on will turn till it faces the sun, and then after a period of time will turn away. This is day and night. In a few hundred of these rotations, the angle of sunlight will change, making it hotter or colder. This is the seasons making up a year. We are born, and our bodies change as they get older. On a given day, I get up, and do this thing now, and then I go and do something else afterwards, until the day is filled up. Meanwhile, all kinds of things are happening in the outside world, and in the universe. Certain people are alive now who weren't in the past, or won't be in the future. What first got me thinking about this, is seeing how the subways and many buildings were in one shape 25-30 years ago, but have been renovated now. When I think of a given year, I think about what everything looked like, and what the culture (music, TV, what was in style, etc) was like, as well as what school or job I was in, and where I lived. Stars have become novas, and the planets line up in different ways. We have sent spacecraft to them, and beyond. All of this is also MATTER. Even our perception of these things comes from our brains; also matter. So time is marked by circumstances (states of matter), or events (changes in matter or energy). So time can be defined as an imaginary medium in which the state of or changes in matter is measured. (It can be considered imaginary because you can only be in one point at that given time; you are not really "moving through" anything). This too is basically local, because at different speeds, and in or out of gravity fields it has been shown that the changes to matter (natural processes or even our perception) occur at different rates. So for this reason, time appears to move at different speeds, or even stop completely in certain extreme cases (light speed, black hole event horizon).
TIME TRAVEL?
Because time has been observed slowing down or speeding up, and we seem to move through it like we do through space, many feel why can't we turn around and go back in it? But now with our definition of time as a[n imaginary] medium in which change of matter is measured, going back in time means reversing all changes to every particle of matter around us. For something that was burned up, each carbonized particle must be un-carbonized and come back together as it was at the given time you want to return to. Someone who died must come back to life, his body un-decay and get younger. Someone born must grow down and return to an unfertilized egg and separate sperm cell. Everything else in the universe must simultaneously revert to prior states. Perhaps you could localize it and only make everything in one room revert, but as soon as you stepped out of that room, it would be the present again. What point in time would it really be? (This would be analogous to twisting or bending time).
And it seems then, that one person could not even go back in time by himself (if on a universal scale). Everything would go backwards and everyone with it. Their consciousnesses too (since that is a material process)! So perhaps, no one could even perceive a reversal of time. Unless the one traveling exempted himself from the reversing process; remaining in the present of his own consciousness. So actually, that person is not traveling in time, but actually sending everyone else; the entire universe around him, back in time. (What would happen to stuff he changed? It would still probably un-change, but it would appear to be a ghost where he was if he doesn't un-enact everything he did. If he could then restart everything forward again, perhaps he could alter the past. But what would happen to everyone else's consciousness? I'm not exactly sure.) But it does not seem possible to for someone to reverse the entire universe around him, so it does not seem that going back in time is possible. What are you going "back" to? Just a particular set of simultaneous states of matter that have since been changed. It's not even really "back"; as if it were a location in space that still exists and thus could be returned to. Existence is only in one point of time. No-one knows how to recreate this simultaneous state of all matter, since matter constantly changes and no two states of it are ever exactly alike (just like no two pieces of matter are ever just alike). And then you have the fact that even "simultaneity" is reckoned differently to different observers at different speeds, as General Relativity teaches. So again, what would even determine a "simultaneous" point of the "past"?
Scientists have theorized possible ways to return to the past, such as wormholes, and something about a rotating mass of dust spanning the universe (?); but these are all conjectural and un-provable.
MULTIDIMENSIONAL TIME?
The easiest way to conceive of two-dimensional time is to picture a bunch of people living and moving about in an area filled with statues. But these statues are really the inhabitants of a perpendicular (2nd) time dimension. Our whole time line lies only in an instant of theirs, so we see them frozen in an instant of time; and their whole time line lies in an instant of ours. So we are frozen in time to them. Now how could we appear frozen to each other? Well, if you could use a stopwatch (like in Twilight Zone story "A Kind of Stopwatch", or the made for syndicated TV movie The Girl, the Gold Watch and Everything, and it happened automatically without a stopwatch in Stephen King's "the Langoliers"), and leave our time, and enter theirs, then everything in this world would stop, and the frozen statues of that world would start up, moving around. If you stopped the watch, then that world would stop, and this one would start again, all the objects in this time dimension picking up exactly where you left off. But now all the people in the 2nd time dimension have changed position from where you left off. So you're really not back in our time dimension, but in a parallel time dimension to ours, in a different position in the 2nd time dimension. From this dimension, you would disappear; going into hypertime is pretty much like disappearing into hyperspace. You could travel through both dimensions at the same time, in which you would see both times running in slow motion. With the right trajectory, you could reach the same point described above by stopping one time, living completely in another, and entering the parallel-to-the-first time. If you change your path to a smaller angle to one of the time lines, that one speeds up, and the other one slows down.
Now all of this could be taking place amidst other statues which are inhabitants of a third time dimension, and you could carry it to any number of dimensions, getting more complex as you increase the number, just like with space dimensions.
Where this might fail, is that the most popular definition of time, is in terms of the causality of events. Events only have one cause. Two dimensional time would be defined with events having two causes. And 3D time would have events with 3 causes. Multidimensional events are not clearly defined in the scenario I've given. Instead, it is more like our time, where if you approach the speed of light relative to earth, then the other frame of reference's proper time will stop. One Scientific American article on the subject a few years ago ("Parallel Universes", Max Tegmark, 5-2003, p.45 http://www.scribd.com/doc/17662852/Parallel-Universes), had a table with several space and time dimension number combinations, and only 3D space/1D time could sustain reality as we know it. Less dimensions of space, "complex structures cannot exist". 0-D space and/or time, a red backgrounded "events are completely unpredictable". More dimensions of space "atoms are unstable". 2 or more dimensions of space with more dimensions of time together, had a green-backgrounded "events are completely unpredictable". I wonder if the different color meant it was not as bad as the 0-D scenarios (which obviously could not have any events at all). The flipside of our universe: 3D time/1D space, had "fields are unstable". I don’t know what that means, but it is funny that that does not seem to be as restricted as the other combinations. So complex structures can hypothetically exist in that 1D space, unlike ones with less time dimensions? Likewise, 1D space with 4 or 5 dimensions of time also had "atoms are unstable", like the reverse. I wonder what would those universes be like. Why would three or more dimensions of time only be viable with one dimension of space, and not more? Who knows. And who says such "universes" would even have the same laws ("the standard Model"), which governs things such as atomic stability? With different laws; atoms might be stable with more dimensions! (And Michio Kaku and others have theorized, apart of string theory, on other 3D space/1D time universes like this one that might have other than Standard Model laws, including a cataclysmic condition where the laws in this universe could change from the standard model, and matter would all break down and reform!)
Here, btw, is a great way of explaining the fourth dimension, from scratch, for beginners: Fourth Dimension: Tetraspace
Another good explanation of the fourth dimension: "A Study of Dimensions" by Bill Price. And a video building up to higher dimensions in a similar fashion is at the bottom.
Explanation of "bending" of time
Micho Kaku's Hyperspace: A scientific Odyssey of Parallel Universes, Time Warps and the 10th Dimension (NY, Oxford Press, 1994) renewed interest in alot of this stuff for me. (I first got into it through Carl Sagan's Cosmos series, which among other things, explained the fourth dimension, and it's hypercube, the "tesseract", by comparing it to how the second dimension relates to our space. This is instrumental in trying to understand higher dimensions).
One curiosity Kaku mentioned, to try to explain why time supposedly slows down when you approach the speed of light, is an imaginary scenario where the speed of light was only 30mph; the speed a subway train enters the station. (p.84; note, p.341). It would appear compressed down to maybe about 1 foot long, yet to the people on the train, it would be the station that appears compressed to one foot long. The foot would expand to the full 600ft as the train slowed down. This seems paradoxical, as how can both be shorter than the other? But he points out that it takes time to do the measurement. So if a person on the platform takes out a foot long ruler, it will measure the entire train. Yet if he drops the ruler onto the platform so that both ends hit the platform simultaneously as the train passes by, the people on the train will not see the ends hit the ground simultaneously. One end of the (compressed) ruler hits the floor first as the front of the train goes by, and the other end hits as the station is completely past. So this is how the paradox of a one foot ruler measuring a 600 ft train is resolved. But now it rasises another problem.
If the ruler is dropped so that both ends touch the ground at the same time, then it is always parallel to the ground. So how do the observers on the train see one side touch the ground first —i.e. it is apparently not parallel with the ground? Well, the compressed ruler is seen in the compressed station standing up on one end, so it measures 1/600 ft horizontally, but one foot vertically. The 1 inch mark is one inch off the ground; the 2 inch mark is 2 inches off the ground; the other end is one foot off the ground. How does this happen when the entire ruler was the same distances off the ground to the other observer? Well, think of it this way: was the one inch marker ever one inch off the ground? Yes; it had to pass that height when the entire ruler passed that point right before it hit the ground. Was the 2 inch mark ever 2 inches off the ground? Yes; right before it was one inch; when the entire ruler (including the 2 inch mark) passed a two inch height off the ground. Was the other end of the ruler ever one foot off the ground? Yes, even more time before that, when the entire ruler passed that height of one foot off the ground.
So do you see what is happening? The observers on the train are witnessing the actual bending of time. They are seeing each point of the ruler at a different time, when it was at different heights from the ground! (Going further into the past the further towards the rear you go). Likewise, the observers in the station see the rear of the train in the present; what we can call event B; but the front of the train apparently one foot ahead is actually being seen at event A, in the past; because the front is really 600 feet ahead, having already passed the one foot point. So the train is compressed because each point on it is being seen at different points in its own proper time, going further into the past, the further to the front you go.
My reason for discussing this, is not only to draw further on Kaku's illustration; but also to illustrate the supposed problem with faster than light travel, as also discussed by Kaku, Rudy Rucker's similar book The Fourth Dimension, and especially William Kaufmann's The Cosmic Frontiers of General Relativity. If time supposedly stops at the speed of light, then you may think it would go backwards, faster than the speed of light. Some have created stories, and even a poem based on this. But it is not necessarily true. If you headed towards Alpha Centauri at four times the speed of light, it would still take a whole year on earth's "coordinate time" (the time of observers who maintain the same relative velocity from the start of the observation, or in their own frame of reference, remain "at rest") to get there. You then turn around and return at the same speed, and your total journey would still be two years. Time would not become negative, because the equations governing this phenomenon include square roots; so when one factor falls below 0, you end up with not negative numbers, but rather square roots of negative numbers, which are known as imaginary numbers. That is a fitting description of FTL mass, length and time when compared between proper and coordinate time! Something moving AT the speed of light supposedly would be like infinite speed. You could go anywhere in the universe instantly in your proper time, though it would be billions of years in coordinate time. What we would normally consider infinite speed, like the "Kind of Stopwatch" example, would be well into the FTL range (of course), and cause problems of causality, even though you would THINK you were still not going into the past, as shall be explained. It seems like some sort of disconnect between proper and coordinate time. A very finite speed of 670,616,629 miles per hour becomes like infinite speed in several respects. You would think light would move at infinite speed. It is not even light itself that determines this speed. It's that light has no mass, and any particle having no mass will normally move at the fastest possible velocity, and this for some reason is not instant to the rest of the universe, though it is to the one traveling at that speed. I have no idea what the FTL traveler's "imaginary [proper] time" would be like, as we watched him make the two year round trip in our coordinate time.
The problem with the 4c trip occurs if on the way out, you "catch up to" another astronaut, heading that same direction just under the speed of light. Instead of floating around Alpha Centauri, you beam aboard this other ship, and ride with him for a couple of seconds, and then try to return to earth at the same FTL speed, from his frame of reference! With the "bent time subway station" illustration; I can now easily show what happens. Let's say people on the train and station try to fire a tachyon (a hypothetical particle that can only move faster than light) to each other. This was conceived to get around the issue that we could never accelerate to the speed of light, let alone beyond it, because of the time contraction (as well as mass and required energy increasing as well). So they might have some sort of receptors that emit and register its reception with an indicator light or something, since they could not actually "hold", "throw" or "catch" such a particle at velocities less than light. If the person in the station at event A passes the tachyon to someone at the front of the train as it passes by him, and then that person passes it to someone at the rear of the train (at the same speed. Perhaps infinite speed, even, or it can be any finite multiple of the speed of light), and then that person tries to fire it at back to the station, what happens? When event A is seen by the front of the train, event B hasn't happened yet!!. He sees the back end of the ruler still off the ground, like before the train passed it. And this makes sense, because to the observer at the front of the train, the back of the train is not 1 foot, or 1/600 ft back (the distance of the back end of the ruler from the front end) — it's 600 ft back! — not even in the station yet! From the viewpoint of the simultaneous event A-B in the station, it is in the past. The tachyon, leaving the back of the train, which is not in the station yet, will get to the station before the back of the train, and events A and B! It jumped into the past and turned up before it was originally even omitted!
Also, when the back of the train reaches event B, from that frame of reference, the front of the train is not 1 foot or 1/600 ft ahead, it is 600 feet ahead —already past the station. So from one viewpoint, when the tachyon is first passed from the station to the train at event A, it is jumping into the past (because the station sees event B simultaneously with event A, yet from event B, the front of the train was really past the station already, and thus only in the station in the past). From the other viewpoint, it moves into the past going from the front of the train to the back.
So basically, scientists conclude that if tachyons exist, they do not interact with the familiar "tardyon" matter of the visible universe. So a receptor would never even be able to register receipt of a tachyon. They all share the same space, but have the space and time coordinates sort of reversed from each other. I wonder if there might be some other "cosmic censorship" principle that might prevent that scenario. Like spacetime itself "remembers" where/when you or the tachyon originated, and your "world line" (projection of locations in space through time generated as a path through "space-time") simply would never be allowed to cross itself, (as would happen if you went back in time, returned to your starting place, and then resumed normal time), and you would simply appear at your starting point at the next allowable time after you left.
P Theory/F Theory (Patrix/Father): The Third Continuum
Some theorists have speculated on alternate realities. Particularly when the subject of reverse time travel comes up. If you go back into the past, and begin altering reality, causality is violated, and the universe would make no sense. So they speculate that if you do go back into the past, you would in fact enter a parallel spacetime universe in which events are different. Each instant in time, when an event occurs, such as when we make one decision or another, becomes like a "fork" where multiple realities split off, and only one becomes realized in our experience; the one "chosen" over the others.
So there is a notion that the other, unchosen realities ("counterfactuals") might exist in some way, yet we simply don't have access to them. If you accept that the universe and all its matter, events and space and time itself consists of vibrating loops of string, then it doesn't seem farfetched to believe that all of these other states of them (collective vibrational patterns) exist, yet only one path through them has been chosen. I have seen two articles, using both space and time to access these counterfactuals. One, in Scientific American, proposes an infinite space, and the further out you go, you run out of possible configurations of matter and energy. So all matter can do is to start repeating itself. Eventually you will run across exact and near copies of everything we see around us, including ourselves.
The other one (http://www.exitmundi.nl/eternity.htm) does the same thing using infinite time. After the entire universe burns itself out and cools into nothingness, the quantum uncertainty principle proposes matter will be randomly popping in and out of existence. Given enough time, more complex objects will appear, including eventually, copies of everything we see around us, including ourselves and a new big bang.
One problem I have with infinite space or time, is that matter and events become zero in the overall scale of the universe.
But to me, parallel realities created from being "unchosen" in the here and now (and not cast off into infinite distance or future simply because the universe has run out of possible combinations) creates a new continuum, in addition to space and time. Space is the medium in which we measure the relative locations of matter and events. We use it to get from one location (marked by matter or an event) to another (see above). Time is the medium in which we measure the chain of causality or simultaneity between events. We use it as we live and experience one event after another.
So think, what medium would be the one "travelled" by jumping straight from the here and now in our universe, to a parallel universe where we wore red instead of blue in the here and now? You might think time, but time is marked by a causal chain. On event causes the other. Yet an event we are experiencing now did not cause an alternate event in a parallel universe. They are results of a different choice (event) at a point further back in time where the two realities merge. The causal chain lies in the forward time dimension itself, not in the perpedicular dimension in which you jump from one to another.
It is also not space. When we think of "parallel universes", we are usually thinking of space, where one space is embedded in another space with more dimensions, containing other lower dimensional spaces parallel to the first one. You might think that alternate realities would be the temporal counterpart to embedding in higher spaces. But again, in higher dimensions, the medium in which the lower spaces ("branes") are located relative to each other is still space. The medium between counterfactuals is not time, because, once again, the relationship between parallel corresponding events is not causal.
So this is an all new medium. To give it a familiar monosyllabic name like "space" and "time", I would call it "chance".
I always like things like this in threes. So we have space, time and chance. It sort of parallels the Christian concept of the Trinity. And creationist Henry Morris and others have even linked the concept. While the Trinity is often thought of as three equal beings sitting side by side, Morris framed them in terms of a reference, a visual form and an experential form. So the Father is what God is, or who we reference when speaking of God. The Son is God made flesh, visible and tangible in the world (the Father cannot be contained in space and time). The Holy Spirit is how God is experienced (in the heart). Morris ultimately still holds the "traditional" symmetrical view of three "equals" side by side. But when I researched all of this, I found that the pre-Nicene church fathers actually held a non-symmetrical view in which the Father was the Godhead from eternity, and the Son and Spirit were manifested from Him in time (i.e. at the birth of Christ). Forms of this were later revived by the likes of Marcellus and Servetus, but the church by then condemned them in favor of the symmetrical view, which has become the official dogma ever since. (more on this at Triune.html)
Morris had linked this tri-unity to the universe, which he said was referenced to space, seen in matter and experienced through time. (and space had its three dimensions; and time had past, present and future, etc).
I did not like making space the "Father-like" element and matter the "Son-like" element. I had already started coming to see space and time as the visible/experential counterparts, and expected the third continuum to be another kind of continuum like space and time, unlike matter. Matter is what occupies space, and that may appear to fit the "visible manifestation" role, but it is not the same sort of thing. (Though mass is often the third measurement next to distance and duration in equations). You can imagine a universe without matter. Measurement of distance and events then becomes irrelevant, but it is still hypothetically possible. But a universe with no space or time is a whole different kind of existence.
So if we look at the universe as the entire set of possibilities, then this new "chance" continuum is what it is "referenced" to, and space and time are manifestations of it, with space as what it is seen in and time what it is experienced in. This makes sense, as space tells you "where" and time tells you "when", but chance tells you "WHAT", in the first place!
Just as you can get "close" to a point in space (with gradual changes as you pass one material object after another. Think of the transition from country to city as you get closer and the density of people and buildings gradually increases), and closer to a point in time (as one event leads to another, and a new "present" takes shape. Think of any transformation in time. A flower or other living thing growing, etc), you can also hypothetically/theoretically move closer to a point in chance by changing things to alternate states.
Like if I chart my position using the four dimensions of spacetime. At such and such time, I am at a particular longitude, latitude and altitude. In an alternate reality, I may have moved to a point five feet away in latitude. Or maybe ten feet away. That would indicate a further "distance" in chance, since five feet away is "closer" to the starting reality than ten feet away. Of course this will affect the choices I would have had to have made in space and time, in order to get to that point. Just like space and time determine where you can go in each other. So this is interchangeable, just like space and time, and can also be measured. While space has three dimensions, and time has only one dimension, the number of dimensions in chance seems to be unlimited.
I've recently been thinking more on this, and trying to come up with even more fundamental definitions.
I would say that with strings as the fundamental fabric of space-time, any given point in time is a collection of string-vibrations (i.e. all the strings making up space). Some strings are just empty space-time. Some are gravitational fields, some are matter, and some are energy. They are all arranged a particular way at any given time, measured by relative location in the medium called "space".
The entire "matrix" is a set of all possible arrangements of string patterns, yet only certain arrangements are "actualized". "Events" are changes in matter and energy (either changing form, or changing location in relative to other events). Space, time and chance are simply the continuums in which events are located relative to one another. This can be space, time or chance (which would be counterfactual events, or "alternate reality", which are simply the possible events not actualized).
The way to understand this, is to imagine the event of the beginning of the universe (supposedly expanded out from a single point). Now, the string pattern begins changing, and one pattern leads to another in a causative chain. First, matter is compact; then it has moved further apart. There are three coordinates in which it moves, and a different arrangement of strings can be found.
So there are actually four paths connecting to different string arrangements. One is causative, and you can only experience events in a chain once. The other three consist of all the displaced chains of causation, where matter and energy has taken other shapes in other events, yet these chains can be accessed from one another. I say chains and not events, because you cannot travel between simultaneous events, because that is faster than light, and even simultaneity is relative! So instead, you can access another event that stems from the event you observed from a distance. Hence, a different "chain". So each piece of matter you look at in space is apart of another chain of events (its creation, changes made to it, things done with it, etc), displaced by whatever "distance" you measure, from the chain of events you are apart of.
Each collective pattern arrangement is actualized when all the chains of events leads to that particular pattern. (You scratch your head, as the object you are looking at reflects a flash of light from outside, and everything else occurring at that time).
So the continuum along which the chains progress becomes causative, and called "time", and the continuum (with its three coordinates) in which all the different chains are simultaneously displaced is called "space", and we have freedom to move in any direction. Chance would be the hypothetical continuum connecting to the un-actualized arrangements.
Like if I wear a red shirt, that is one pattern of strings actualized at the same time as every other event occurring in the universe. But a parallel timeline where I wore a blue shirt instead could be thought of as possible, and even "existing" in some hypothetical way, but it is simply not actualized. (Putting on a blue shirt after the red one doesn't count as actualizing, because then in that time frame, everything else in the universe has progressed (changed) from what it was at the time of the initial event, so it would be a yet different arrangement).
You can see this stuff in theoretical physics' books, but I have never seen any of them speak of this continuum in which parallel timelines exist, as a third continuum. It's not time, as there is no causal chain between parallel Time and space are frequently said to be interchangeable, and I would say chance is interchangeable with them. All they are is three different methods of arriving to different potential universal collections of states of matter and energy. One of them you have free movement in, another you are involuntarily "dragged" along, and the other you have no known access to at all. In this idea, chance is basically the primary continuum of the universe; in which are defined the connections between all possible events or collective states of matter and energy. Space and time are generated from it depending on the relationshipe of events that have been actualized.
Space is the continuum of the connections between actualized events that exist independently of each other.
Time is the continuum of the connections between actualized events that generate (cause, lead to) one another.
To further elaborate on something said before; an alternate causal chain springing from an event is not time because it is not actualized, and we are defining the continuum of "time" as causal connection between actualized events. So such a chain would simply be a veering off into this new "chance" continuum. Yet if we could actualize that alternate chain of events (at the same time as the first chain), then it would sweep out a causal plane rather than line, and thus generate a second dimension of time. So in this, a chance dimension has transformed into time.
If we start with an old neighborhood, we have our coordinates of how far east or west (longitude), north or south (lattitude), and of course that it is on the ground (up or down; altitude), and then choose a particular time, then we have an event with at least five coordinates; the first three telling you where, a fourth telling you when, and a fifth (and more) telling you WHAT: a particular state of matter.
After this initial event, the neighborhood can either have its buildings stay the same, or be renovated, or it can be torn down for an all new development. Either way, we will arrive at one of two different possible events, separated from the starting event by a causal chain, which will replace the starting event with the new event, eventually. Only one of the events will be actualized. The other will then lie outside of the four dimensional spacetime continuum, in the "chance" dimension, and we will say it doesn't exist.
Now, if we had a dimension in which we could actually lay out the entire causal chain (world lines springing from both events) along it, both events could be seen at the same time. Both would be actualized, and the old neighborhood and new neighborhood would exist side by side in the same spacetime continuum, yet occupying the same lattitude, longitude, altitude and time. You could now freely go back and forth between the old neighborhood and the new one. New causal chains could then spring forth from both that do not replace one with the other. (Like the old buildings being renovated). The displacement between them would no longer be causal. So then, a time dimension has become a fourth space dimension, and a separate time dimension continues to march on.
Here in this video, we see another good step by step buildup of dimensions like the ones linked above. He's building up to ten dimensions, and using time as the fourth, and different kinds of dimensions besides space and time after that. Dimensions 5 and 6 would be the realm I am calling "chance". In the Powerpuff Girls episode, "Get Back Jojo", the professor explains time as the fourth dimension (showing an orthogonal diagram of a tesseract and a mouse in a maze as the lower dimensional analogy), and the dimension used to jump back in time and alter the present/future as the fifth. However, that would really be simply traveling the "other direction" in the fourth dimension, rather than traveling perpendicular to it. If the past is altered, that would then be what branches off a new counterfactual, separated from us by a fifth dimension or "chance" continuum. As this video explains, going back in time is "the long way" to reaching a counterfactual, analogous to traveling around the universe from one point to another, instead of just using the wormhole more directly connecting the two points. Here are some representations of space and time in scifi that I had long ago thought of: The other illustration is that of the "Langoliers" movie. Since that was a kind of 2D time, as was discussed above, yet it added the new twist that these creatures appear and literally eat each instance of spacetime after awhile in the new time dimension; I thought about how that would be represented in a spacetime diagram. A third dimension is added, which is also a kind of time. The plane and people were moving in normal subluminous speed forward in our time, and ahead in space. They suddenly stop moving forward in time. Hence, they can no longer be moving "up" in the diagram, with up normally being used for the time dimension. So now, their proper time is still moving forward in a normal causal chain of events, yet they are frozen in an instant of our time. So they are now moving in the new time dimension, and can still access the three dimensions of space. Soon, the eerie Pac-man like creatures are coming at them, from generally one direction. Let's make it behind them, since they begin fleeing in the direction opposite of where they are coming from, of course. They eat the very fabric of space behind them, leaving blackness. They in essence, are what "clean up" all the "used" instances of time behind them, and then in each succeeding instance of time, a new batch will soon eat the used spacetime. So when we live each instance, it doesn't immediately go away, it lasts a bit in another time dimension, until it is eaten by these creatures. Forgotten 4D object (http://eusebeia.dyndns.org/4d/geom.html): The duo-cone
This page discusses a lot of different 4D objects. One of them is called a "cylindrone", which starts with a cylinder in 3D, which is extruded into the 4th dimension and tapered to a point. A lower dimensional analogy would be taking a circle on a paper, and raising it up into the third dimension (extruding), but at the same time shrinking it to a point as it rises (tapering). This process generates a cone. So if you do this with a 3D cylinder into the fourth dimension, you are creating a kind of hypercone, but with a cylindrical base. So this page had an illustration of this, but the description was originally that of tapering a cone into 4D. That's a different object. That would be a pure hypercone; a 4D cone with a conic base (as opposed to another analogue, a spherone, which tapers a sphere into 4D, which would then be a hypercone with a spherical base).
It's said that if you move at the speed of light, then that direction of space then becomes like time. (I'm not sure if time then becomes like space, since when you move at the speed of light in any one direction, you are still in the others. Within a black hole, space and time are said to switch like that as well, with some sort of freer movement in time, but then you don't have much space or time left, as you are pulled into the singularity and torn apart by the tidal forces).
In part 2 of the video, the 7th dimension connects all the infinite possible "chance" paths of our universe (from big bang to all the possible endings) with all the possible chance paths of entirely separate universes or "infinities". After this, it seems to get a bit redundant, as the 8th dimension is just described as "another line branching off the 7th dimension line to connect to yet another infinity", and the 9th dimension is "all the possible branches for all the possible timelines of all the possible universes". The tenth dimension treats all of this as a single point, but then there is nowhere left to go. The author ties this to the ten dimensions of string theory, yet acknowledging that this imagining is "not the accepted explanation for string theory", in which the other six dimensions are strictly spatial, and collapsed to the Planck length.
However, in that story, since it is seen as having potentially damaging effects on the present, this would represent the other version of time travel theory, where there are no parallel counterfactual universes, and tampering with the past damages causality in this timeline. The "Back To the Future" series is like a combination of this, where changing the past causes things to instantly change in our timeline. So there are no parallel counterfactual universes, however, counterfactual events and aftermaths automatically appear in our universe after a reverse time jump in which something was changed. In the PPG story, it was not revealed whether the negative effects would be instant changes in reality, or just a total breakdown of reality. However, the trip into the past does lead to what is known as a "causal loop" (a self-causing event chain), where the PPG's enemy ends up causing the creation of the girls, rather than erasing them as he intended! The professor had created these girls he remembered rescuing him in the past, which were actually the created girls going into the past to rescue him!
First, the Twilight Zone episode "Little Girl Lost" is describing a hypothetical "tear in the fabric of space" called a "Riemann's Cut". Basically, you end up spat out directly into hyperspace. Hyperspace in this episode was portrayed as a warped, foggy, mirrored realm (with echoing sound) where the notion of a straight line seemed to have no meaning. In reality, in hyperspace, the current three dimensions we are familiar with would be the same. You could go in, jump around and dance and do backflips, and then come right back out, so long as you did not move too far right/left or up/down beyond the diameter of the hole. Even then, just move that far back in those directions, and then head back out, and you'll find the hole. The real problem would occur if you happened to move in the new dimensions[s]! This would have to be involuntary, as you don't even have the muscles to turn towards or move in this dimension! Some force would have to pull you in it.
That force could even be familiar gravity, as one version of string theory says that "gravitons" are loops of string that are free to move on and off of the three-brane of space we live on, while matter and all the rest of the forces have their strings bound to the brane. That would mean, if you could peel yourself off of three-space, you would still be bound by gravity, and begin falling as if you stepped out the window, and we would assume that the floor or the surface of the earth do not extend into hyperspace, so you would continue falling, for thousands of miles, until you reach the spot just 'outside' of the center of the earth in the new dimension, and you would be stuck there with no way to "climb" out. Anything else that fell out of the cut would fall on top of you. Perhaps, if you could move in the new dimension far enough away from the brane, then the graviton field would thin out, and you would float in zero gravity again.
Also, there was the possibility that when the doctor pulled the people out, he could have pulled out skeletons by themselves, or just the flesh by itself, or just skin by itself, all intact, but missing the rest of the body. In hyperspace, every point in the volume of our bodies is directly exposed to the outside, whereas in three dimensions, they are all contained in our bodies. Our bodies are held together in three dimensions, not four or more, so we or any object we are familiar with could come apart easily; perhaps even by just the pressures that hold us together in three-space! In fact, I don't think lower dimensional objects could really exist in higher dimensions, because objects in each n-dimensional space are defined by a "volume" (length in 1D, area in 2D, etc) consisting of the lengths of each of its dimensions multiplied together. If we go into the fourth dimension, that fourth length would be zero, and when multiplied to our height, width and depth, it would all be cancelled out, as anything multiplied by zero is zero! You cease to exist, and all your matter just pops out of existence. Not the normal laws of this universe where matter can't be destroyed! Even if we had a very small hyperthickness, such as the Planck length of the strings we are made of; it would be too "thin" to hold up in the new dimension. So matter wouldn't be destroyed; it would just vaporize into loose individual strings! Like there is nothing that thin with a visible height and width floating around in our space. The only true 2D objects in our space are things like shadows and reflections, and those need 2D surfaces to exist on!
The other universe also could have been simply another three-space attatched to ours at those points. This is really how Bernhard Riemann originally conceived of his "cut locus" of "multiply connected spaces". That universe could have different laws, where matter would not even exist as we know it. Of course, in the story, they speculated on dimensions four or five, but since as we have seen, there are different notions of what "higher dimensions" are, so they may not have necessarily been thinking of what we call "four-space" or "five-space", where tesseracts (hypercubes) or decatesserons (hyperhypercubes) exist.
If it did happen to be five-space, it would basically be the same as four-space. If you are on a line (1D) on a sheet of paper, and you are able to get off the line, you can either jump off to another part of the paper (into 2D), or directly off our the paper into 3-space (basically "jumping two dimensions" as it were).
Since space is, recall, only one dimension in these diagrams, then the Langoliers are going to be coming from one direction (I've made it left), and behind them (to the left, in the space dimension), there is no spacetime left. Even though I have them facing right, which is space-like, they like everyone else moved at less than the speed of light, so their world lines would be less than 45° to the space axis; moving "back" away from us, more than to the right, though faster than the plane, which they were catching up to, as it finally snapped back into our timeline at the last moment! The world line of the "eaten" edge of space would of course follow the Langoliers, and is supposed to be less than 45° but was exaggerated to be different from the second time axis, which is at an angle in this isometric projection.
So I drew this image of the object, which I call a "duo-cone", since it is conic in both the 3D hyperplane, and in the 4D plane. I mailed him about this with the image, and he said he would look into it. He since corrected the description and redid the page with shiny new transparent colored images, but still left off this one. So I might as well host it for now.
When the initial cone begins to be moved into 4D, the circlular base generates a new cone sharing the same base, the apex generates a ridge connecting the apices of both cones, and the sloped, curved sides sweep out a kind of hypertorus that "fills" the figure in the image.