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Universal Law of Life, the Universe and Everything
February 22, 2012, 11:36:03 PM
OK, Lars and DcadRob have been musing about things mathematical, so here is a really mind-numbing problem. And there is a single simple answer.

1. First imagine a series of geometrical objects; point, line (segment), square and cube.

A point is zero dimensional - no length, width or height.

A line segment is the result of moving a point X units in a single direction - one dimensional.

The square is the result of moving the line segment X units in a direction perpendicular to the line - two dimensions.

A cube is the result of moving a square X units in a direction perpendicular to the plane - three dimensions.

A tessarect is the result of moving a cube X units in a direction perpendicular to all surfaces of the cube - four dimensional (proof left to the student, or read "And He Built A Crooked House" by Robert Heinlein).

etc.

2. Each object is bounded by the figure of the previous dimension. Cubes are bounded by squares, squares are bounded by line segments, line segments are bounded by points, etc. All are joined at right angles, making these "right regular objects."

3. Let D be the number of dimensions in an object, and N be the number of dimensions in a part of the object. For a three dimensional cube the bounding figure (square) is two dimensional, so D=3 and N=2. You could also consider the lines in a cube where N=1 and the points where N=0.

4. This table describes the number of N dimensional objects in D dimensional objects:

D \ N 0    1    2    3    4    5
0      1     -    -    -    -    -
1      2     1    -    -    -    -
2      4     4    1    -    -    -
3      8    12   6    1    -    -
4     16   32  24    8    1    -
5      ?     ?    ?    ?     ?    1

For example a cube (D=3) has 8 points (N=0), 12 lines (N=1), 6 squares (N=2) and 1 cube (N=3).

5. Find a single equation that can be used to calculate all relationships of these right regular objects and fill in the blanks for the pentacube (D=5). Don't cheat - figure it out for yourselves. I did. Besides, unless you know the particular branch of mathematical obscura this falls under you will probably never find it, not even with Google.

It took me almost 30 years of searching before I discovered that I wasn't the first to come up with the formula. #^*&%! It was published in 1946! So I know how you feel, Lars.

****

PS: Did you know that an 8-dimensional right regular object has 1792 lines and 1792 cubes?

PPS: OK, so I exaggerated. It really isn't the universal law of Life, the Universe and Everything. But it's about as close as we will ever get.

Phil

« Last Edit: February 22, 2012, 11:38:34 PM by Dr PR »

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February 23, 2012, 06:57:51 AM
Doc, I found a pic of your tesseract on wikipedia (http://en.wikipedia.org/wiki/Tesseract), attached below.

Looking at this, I have to disagree that it's a 4 dimensional object. It's still only 3d. I don't think we can get beyond 3d in a physical sense. Once you get to the cube you can only generate another cube outside it. In theory this would result in a series of pandora boxes with the same amount of points and surfaces connecting the edges of each inner cube. In reality each generation would the result in a regular 6 sided cube since the inner one would simply become the interior solidness of the overall cube. Image pouring concrete around a concrete cube. The result is just a bigger mass of concrete.

Now, if you generate a surface from each point of the cube, each generation would create an ever increasing number of points. I would consider this a different system of propagation entirely, though: a single point (1d) would spawn at least a triangle , since no less than 3 points can define a surface (2d). The 3 points of the triangle would each spawn triangles, creating a pyramid, I think (3d). What will the 4th dimension be? There's a name for it, I'm sure. It's too mind boggling to consider all the possibilities. A single point can conceivably spawn a surface of an infinite number of points and each of those points will in turn spawn a surface of infinite points. No matter how far you take this the result would still be a 3 dimensional shape.

In my humble opinion, and more in line with your topic heading, the 4th dimension is time, the 5th dimension is possibilities (what could have happen during a space in time), the 6th is a story (what actually happened during the time), or something like that... I have a list of 10 dimensions, starting just as you did with the point, line and cube. I posted it on the forum many years ago, before there was a 'totally off-topic' section. Can't remember them all right now. If I find it I'll post it in it's own topic. If you go through the process of creating a 3d animation from scratch your animation will go through each of the 10 dimensions, ending with something like what effect the animation has on those who view it and how it would inspire other animations. In terms of everyday life it boils down to the fact that there is an actual consequence to every action that you can't go back into the past and change but it will affect your future actions.

Lar
ps: on second thought, in my propagation above 1 point can generate 2 points... but you'll never get beyond a 2d shape.
« Last Edit: February 23, 2012, 07:44:58 AM by Lar »

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February 23, 2012, 09:01:46 AM

PPS: OK, so I exaggerated. It really isn't the universal law of Life, the Universe and Everything. But it's about as close as we will ever get.

Phil
[/quote]

Pretty close - missed by .666...

1792/42 = 42.666...

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February 23, 2012, 10:20:11 AM
Lar,

Spoken like a Flatlander! The image you posted is not a tessarect. It is the 3D "shadow" of a tessarect - sort of.

Take a deep breath.

If you make a wire frame cube (real world, using real wire) and hold it between a light and a piece of paper (plane) you will get a 2D shadow of the 3D wireframe cube. In fact, if you create a cube in DesignCAD and view it in wireframe you see this 2D "shadow" on the screen.

The picture you posted is actually a 2D shadow of the 3D shadow a 4D tessarect would project into the 3D universe.

In the "tessarect 1.jpg" picture below you see the "shadow" of a 3D cube (black) on the left. Then there is a stack of 8 colored cubes in a 3D cross shape - Salvidore Dali used this once in a painting. It is an "unfolded" tessarect. Just as you can unfold a cube into 6 squares on a sheet of paper, if unfolded a 4D tessarect would create a group of 8 cubes like this in 3D.

Just as we can fold the 6 2D squares to form a 3D cube, the 8 cubes can (theoretically) be folded into a 4D tessarect. The third figure shows the relationships of the 8 cubes in a tessarect - or what 3D shadows of the 8 cubes would look like. There is the black inner cube, six surrounding colored cubes, and an outer magenta cube. The figure at the right has all the parts assembled into the 3D shadow of a 4D tessarect.

BUT, in 4D all of these objects are cubes, all are the same size, and all are packed within the outer cube (actually, all 8 are "outer," "inner," "side" etc.). If you were standing in the black cube you could pass through any side into a colored cube except the magenta cube. Likewise, when you entered any of the 6 colored cubes surrounding the black cube that colored cube would become the "center" cube and you could access the 6 surrounding cubes but not the new outside cube. For example, if you went from the black cube to the dark blue cube you would no longer be able to access directly the light blue (cyan) cube.

The second image shows these things rendered - except on the third figure from the left I substituted a magenta outline for the surrounding cube.

Is this all gibberish? Well, Heinlein made a great sci-fi story using this "fiction." But mathemeticians figured out the nature of the tessarect long before Heinlein wrote his story.

The book "Flatlands" is an exploration of what the universe would look like to beings who experienced only two dimensions. Their universe would be a plane. But one day the shadow of a 3D cube fell on their universe, and all wondered what it meant. One bright fellow deduced that there was another third dimension and the shadow was the intersection of a 3D object composed of 6 squares - a cube - as it passed through their 2D universe.

Our pathetic ape senses limit us to experiencing only those aspects of the universe we need to "see" in order to feed, mate and survive here on earth. We can't see radio waves or X-rays - or even atoms. But by applying intelligent thought we have learned these things exist.

Likewise, if you try really hard, like the Flatlander you might be able to comprehend the nature of a 4D tessarect by looking at its 3D shadow. Don't be so quick to dismiss additional spatial dimensions as just fantasy. How would you, with your limited 3D senses, go about proving that additional dimensions are not possible?

Phil
« Last Edit: February 27, 2013, 08:32:10 PM by Dr PR »

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February 23, 2012, 04:49:23 PM
Doc, hard as I try I can not conceive of a 4th dimension in a physical sense with time being considered static (as in a picture). I guess that makes me a Threedeelander.

Now I can imagine any number of concepts of a 4th dimension. Let's say the 4th dimension is where the zero thickness of a plane becomes infinite space which you enter by stepping through a hole in the plane. Lets consider size to be a dimension. If we shrink down infinitely smaller and smaller we would pass through different dimensions where our atoms become planets and their atoms become planets. If we expand infinitely bigger we pass into dimensions where our universe become atoms and their universe become atoms.

...Whoa, there, Lar. I think you just hit upon some actual physical ultra-3rd dimensions. Infinite space down, where the atoms in my body become planets in a universe... and infinite space up, where our universe is but atoms in some poor sucker's body or in a speck of dust. Each direction being unending. I mean, to a germ my body is a universe. It's highly probable that my germs have germs.

Is this how A Space Odyssey ends. I read that book in one sitting when I was about 15 and by the end my brain was fried. I had an opportunity to watch the movie on tv but it was so boring that I could not sit through to the end. Ok, what I'm remembering now is that the human passed from adult to baby, signaling a new beginning somehow. He did not pass through dimensions of size. Didn't one of the Men in Black movies ended with the concept that marbles in our realm being planets in another realm. Is there anything new in the infinite dimensions of space?

Lar

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February 23, 2012, 08:05:07 PM
Lar,

No one said time is static in n-dimensional universes. Considering time as a temporal dimension doesn't preclude an infinite number of spatial dimensions. Besides, time isn't constant - it is relative, so there could be an infinite number of temporal dimensions.

Our thought is so biased by our eat-mate-survive imperitive brain that it is very difficult to escape from our mental "Kansas" and consider alternatives. Why 3 dimensions? Well, here on this planet gravity creates up/down, our stereoscopic vision creates front/back and our arms are left/right. Our brains evolved to think in 3D because of these constraints. But would a spherical creature with eyes and arms all around that evolved in space think in 3D?

Considering the vastness of space and time you have to be pretty naive to think the entire universe functions just to satisfy the needs of a wildebeest.

So, back to the problem. What is the equation to describe an infinite dimension universe?

Phil

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DesignCAD user since 1987

February 23, 2012, 08:08:26 PM
cavelamb,

By jove, I think you've got it! Douglas Adams was almost right.

42.332020977...

Call the mice!

Phil

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DesignCAD user since 1987

July 06, 2012, 08:51:45 PM
Hint: 0! = 1

0! is zero factorial.

Factorials are sequential multiplications of n(n-1)(n-2) ... (n-n+1).

So 4! = 4*3*2*1 = 24

Phil

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DesignCAD user since 1987

February 02, 2016, 09:50:41 AM
Four years now, and no one has posted a solution.

Phil

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DesignCAD user since 1987

February 02, 2016, 12:10:27 PM
Have to think about that one.  But on a related subject, I saw a quote the other day,

"What a ripoff  ...  I bought a new Universal Remote  Control, but took it back because I found it did not in fact control the universe  - not even remotely"

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