SOMA News 20 Sep 2001 E-Mail.

# Cubes in 4 dimensions.

Mathematicians often work with the cube, when describing various physical situations, and indeed SOMA was first discovered at a mathematical lecture of space divided in cubes.

One other interesting aspects of the cube is that it is one of the objects of which we can glimpse the fourth dimension.

We will probably never know if the fourth dimension is anything else than the brainchild of mathematicians, but investigating the properties of a fourth dimension makes it a lot easyer for the scientists to describe the way objects interact here in our 3-D world.

When scientists describe the world and the universe in which we live, they often have to take the fourth 'Space-dimension' into account. No one can see this dimension, but just as you can fold a flat two-dimensional piece of paper into a three-dimensional cube, then the mathematicians can compute from our 3-D world, into the 4-D so called 'hyperworld'

The mathematical concept of dimensions is actually quite simple. A dot has no dimension because you cannot move anywhere on it. A straight line has the dimension 1. because you can move in one direction.- (length wise)
Extending the line at a rightangled direction gives us the sheet, like a piece of paper, and here we have 2 dimensions.- (length and width)
Accordingly, we can extend the sheet in a direction perpendicular to the sheet surface in order to get the cube, now with 3 dimensions.- (length, width and height)

Now, if we take this a step further. Extending the cube in a direction perpendicular to ALL the existing axes then we enter the hyper space having 4 dimensions.

Let us see how dimensions govern the evolution of a 3-D cube, how we may view the shadow of a 4 dimensional cube, and how a 4 dimensional cube will look when it is "unfolded" to our 3-D world.

• A point has 1 terminal point (By definition).
• Moving a point in a straight line produces a Line with 2 terminal Points (Corners).
• Moving the line along a straight path produces a Square with 4 corners.

The numbers 1, 2, 4, are in a Geometrical Progression where the next number is then 8.
And indeed, moving the square along a straight path produces a cube with 8 corners.
A logic assumption is then that moving a cube along a straight path will produce a hypercube with 16 corners. - And so it does.

From this we may deduce the numbers that a 4 dimensional equivalent of a cube, will have:

 Shape Dimensions Corners Edges Faces Volumes Dot 0 1 0 0 0 Line 1 (2) 2 0 0 Sheet 2 4 4 2 0 Cube 3 8 12 6 1 Hypercube 4 16 32 24 8

It is difficult for us to imagine that this should be possible so let us back up for a short while, and unfold the cube.

A 3-D cube consist of 6 squares of 2 dimensions.

We all recognice the familiar 'cross' shape, that can be cut out of a sheet of 2-D paper and then be folded back into a hollow cube.
Likewise it is possible to "unfold" a hollow 4 dimensional hyper cube.

A Hypercube also have a - sort of - surface, consisting of 8 cubes of 3 dimensions.

So doing the unfold, we get a 3 dimensional structure resembling a 4 armed cross.
Still - although difficult to imagine - this figure can be cut from solid 3-D materials and folded back into the 4-D hyper cube.

Now, if we look at the familiar 2-D cross, we can recognise the middle piece as being the bottom of the finished cube, and the 'tail square' as being the top side of the cube.
In a quite similar fashion we must imagine the center piece of the 3-D"cross" as being the "bottom" of our hyper cube, and the "tail piece" as being the "top".

Although we cannot see this world of 4 dimensions, mathematicians can tell us, that strange things can be done there.
For example: Rotating a 3-D figure through a 4-D space and back will produce a mirrored shape. Just like you do, if you write your name on a piece of flat glass, and then flip it through our 3-D space, to get the underside upwards.
So sending your "left shoe" into the 4-D world, rotating it, and getting it back, will produce a "right shoe"

In the 4-D universe the SOMA pieces #5 and #6 are no longer different pieces. They are identical!
To a 4-D person, SOMA pieces are as flat as TETRIS pieces are to us.

On the chemical scale, sugar will taste like starch and starch will taste like sugar. Because the molecules are 3-D mirror's of each other.

 To read more about this analogy, you might read the book about Mr. A. Square. living in Flatland written by Edwin Abbott Abbott. http://www.chestnutcafe.com/cafe/index.html?flatland or http://www.malaspina.com/etext/flat01.htm The complete book: Flatland: A romance of many dimensions http://www.geom.umn.edu/~banchoff/Flatland/

Just as we can see the interior of a figure drawn on a flat 2-D piece of paper, so a 4-D person can see the interior of all our objects.
The 4-D person could remove anything from a closed drawer, without opening it, simply because our drawer is closed on ALL 3 sides, but not on the 4-D sides, leaving free access to all contents.

Now, HOW would a 4-D cube look to us, if we were standing in front of it.?

• ### Three-dimensional slicing

To answer such a question, we must first examine the consept of intersections.

Imagine a 2-D creature living on a flat surface. and imagine further that a 3-D cube is about to fall through his world.
Now. what would he see ?
Well. it depend on how the cube is orientated.!

• If the cube falls flat on the surface, he would see a square that appear and disappear.
• If the cube falls on an edge, he would see a line that would grow into a square, and thereafter diminish and disappear.
• Finally. If the cube falls on a corner, Like the image to the right, he would see a spot, growing into a triangle, then into a hexagon, back into a diminishing triangle, and finally to disappear in a spot.
A strange experience this would be for the 2-D creature.

Click this figure to try the intersections yourself.
• ### Four-dimensional slicing

Quite likewise our experience of a 4-D object will be a 3 dimensional object of complex appearence. By slicing our 4-D hypercube into the 3-D sections that appear in our 3-D world, we can gain some insight into the 4'th dimension.
It does require a LOT of imagination, because like the 2-D creature should imagine our 3-D cube by viewing its slices, so we must imagine an unseen 4-D figure, but viewing it's 3-D slices.

Like it is for the 3-D cube, the orientation of the hypercube does influence what we see.
In fact, the slices of the hypercube very much resemble those of the cube. For example, if a hypercube passes through our space head-on (starting with one of its eight three-dimensional cubic faces) then it looks like a series of cubes, all equal in size, just like slicing the cube face first gave a series of squares.
Do you see the parallels?

Slicing the hypercube from a two-dimensional side first gives resuls analogue to the cubic slices from an edge.

Now visualize the shape of the hypercube by slicing it from a vertex see the figure to the right (Imagine it, being a 4-D hypercube, suspended in a rubber string, dangling up and down through our 3-D world.)
To try the slicing yourself, click below.

Click this figure to try the intersections yourself.
• ### Fold-Outs

Slicing is a good way to understand shapes better because it breaks them down into a series of lower-dimensional objects. Another good way to understand a shape is to try to build it from a lower-dimensional "fold-out," like a cardboard box unfolded and placed flat on the floor.

• ### Three dimensional fold-outs

Consider this picture of an unfolded cube:

Does it look like a cube? Now, if we were to give you instructions that said "connect the blue edges" to build a cube? The best a 2-D person could do would be to stretch the squares and align the blue edges together in the plane.

But this was not what we were looking for. We wanted them to fold up without stretching the squares, something that requires three dimensions, something they would not be able to do. The fold-out and instructions for putting it together could nonetheless help the 2-D people gain a better appreciation for what a real cube is.

Given a third dimension to work with, the foldout could actually create the cube by connecting the blue edges.

• ### Four-dimensional fold-outs

For us, who live in a 3-D world, we can gain a better understanding of four-dimensional shapes by examining their three-dimensional fold-outs.

Imagine a "cube" where 3 sides are missing, like the drawing above - the minimum material required to build such a cube, is three squares. When we talk about the fourth dimension, the situation is analogue:
The minimum material required to build a 4-D shape is three cubes arranged around an edge, and then connecting the blue faces without stretching the cubes.

As 3-D persons we cannot complete this task. The best we can do is to stretch the cubes to align the blue faces.

But a four-dimensional creature could fold the figure into a hypercube with no problem.

Here is the complete hypercube fold-out:

As mentioned before, notice here the similarities of this fold-out to the cube fold-out.
We have a central object (square or cube) surrounded on each face with other identical objects, and then a final one stuck on the bottom.
It is easy to think of the center object as being the "base" of the folded-up figure, and the extra object on the end as being the "top" of the figure.

### Now - Try to rotate a hypercube yourself.

One thing is to read a static text about a subject, especially about such a graphically intensive subject as 4 dimensions. Another thing is HANDS ON experience.
To allow just that, Michael Gibbs has written a very impressive Java Applet that shows 4-dimensional polytopes, (Figure) and lets you rotate them.

And the personal homepage of Michael and Tina Gibbs is: members.aol.com/jmtsgibbs/
Email(home): JMTSGibbs@aol.com

Here is an alternative link to the same Applet: www.innerx.net/personal/tsmith/Draw4DApplet.html

• ### Here is a theory of my friend 'Courtney McFarren'

When we talk about mathematics, my good friend Courtney always starts talking - a lot - which is great, and when the talk then turns 4 dimensional he has some very interesting ideas (and I believe he is right)
Here's what he said:

If it wasn't for 4-D space, gravity would not exist. Our universe is 3-D space wrapped around a huge 4-D hyper-sphere. All matter is attracted to each other, so they push towards each other and towards the center of the hyper-sphere, leaving "space dents" around them. These dents (or pockets) warp the space around any piece of matter, creating the "hold" of gravity. The larger the piece of matter, the deeper the "dent" becomes, therefore the stronger the gravity.

When we view a distant galaxy, not only are we viewing as it was billions of years ago, but we are also viewing it when the radius of the universe was much smaller. Because of that, the light that reaches Earth had to travel in a long "spiral" path, but not a short "arc" path, which is probably why we think certain objects are farther away and older than they really are.

• ### Finally - Here are a few other interesting Math links.

http://www.sciencenews.org/index.asp Science News magazine

http://www.forum.swarthmore.edu/mam/00/612/people/abbott/index.html Themes in math relating to 2-D space

Edited by Thorleif Bundgaard.

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