The 4th Dimension
We’ve already discussed the idea of dimensionality. How taking a one
dimensional line segment and moving it in a direction perpendicular to every
part gives you a two dimensional square. Then moving the square in a
direction perpendicular to every part of it gives you a three dimensional cube.
In Edwin A Abbott’s story “Flatland”, the square experiences, and then tries to
teach, the idea of “upward, but not northward”. It is easy for us to understand
this concept that the square has such difficulty with because we can perceive
the third dimension. What is the next dimension for us? What is our “upward,
but not northward”?
Let’s talk about some related concepts. Take a one dimensional straight
line segment bounded by endpoints A & B:
A
B
It is impossible to get A & B to meet without leaving the dimensionality of
the line. To get them to meet, you would need to enter the second dimension,
the plane. The simplest form of this would be a circle:
AB
Okay, let’s try this with a two dimensional object. Now take a two
dimensional rectangle, bounded by two pairs of edges. If you wanted to get one
pair of edges to meet, you would need to leave the two dimensions that the
rectangle exists in. Connecting the edges would give you a cylinder.
A
B
A BB
Now… what about a cube? It’s bounded by 3 pairs of opposite faces.
How can you get a pair of opposite sides to touch each other? You would need
to leave the 3rd dimension to achieve this. Much in the same way as you had to
leave the given dimension of each of the above examples.
A
B
Now, let’s consider a circle and a point outside it. How can we get the
point inside the circle without breaking the circle? You would need to pick the
point off the page, and move it inside the circle- entering the third dimension.
Similarly, to get a ball inside a hollow sphere without piercing the sphere, you
would need to enter the fourth dimension. In the late 1800’s physicist and
astronomer J.C.F Zollner claimed to have proven the existence of the fourth
dimension. His proof was based on experiments conducted with the American
medium Henry Slade. Zollner claims that in several experiments done with
Slade, Slade was able to untie a knotted cord whose ends were fused together,
without ever touching the cord. Additionally, he claimed that Slade was able to
join solid rings together, transport objects out of sealed three dimensional
containers, and write on paper placed between two slates sealed together.
Have you ever accidentally put the left glove on your right hand, or vice
versa? It doesn’t fit, does it? The finger holes are out of line. Now, I’ve been
stuck before and put the glove on backwards just to make it work, but that’s
not what we’re talking about here. There is a symmetry there that makes it
seem like it should be able to work. The gloves are reflections of each other.
Let’s compare this to a simpler example. Take two shapes that are reflected
images of each other in the plane. How would you get them to fit on each other
perfectly? You would need to rotate the shape through the third dimension,
over the line of symmetry onto the 2nd shape.
If you were able to move through the fourth dimension, you would be
able to fit your right hand in your left glove analogously. Difficult to imagine, I
know. Perception of the 4th dimension is seemingly out of our grasp. But that
doesn’t keep it from existing or, ultimately, being possible.
So, that’ll temporarily suffice for our discussion on the concept of the
fourth dimension. There’s so much out there about it, but we must move on.
As there are polygons in two dimensions, and polyhedra in three, there are 4
dimensional polytopes as well (polytopes being the general term for all
dimensions). There are 6 regular polytopes. They are:
1. the Hypercube or Teseract, constructed from eight cubes meeting
three per edge.
2. the Simplex, constructed from five tetrahedra, three per edge
3. the 16-cell, constructed from sixteen tetrahedra, four at an edge
4. the 24-cell, constructed from 25 octahedra, three per edge
5. the 120-cell, constructed from 120 dodecaheda, with three per edge
6. the 600-cell, constructed from 600 tetrahedra, five per edge
Since we can’t visualize four dimensions, the only way that we can
perceive these is by “unfolding” them into three dimensional nets or by viewing
projections of these shapes into the 3rd dimension. The nets work much the
same way as 2 dimensional nets of three dimensional objects work. You can
“cut” the four dimensional shape and “unfold” it. Below is a sequence of
unfolding a hypercube taken from www.geom.uiuc.edu. Notice a cube “pop out”
with each step.
The second method of perceiving four dimensional objects is that of
projections. Some people may not understand what a projection is. There are
two ways to analogize a projection. The first is to think of them like shadow
puppets. If you were to put a flashlight behind an object, what would its image
look like against a wall? Let try this with a cube:
If you were to put a flashlight behind a cube you would get very different
projections, dependant upon the orientation of the flashlight and cube. For
example, if you were to have one face of the cube facing the light, the projection
would be a square:
But if you were to orient the cube such that one of the vertices was
facing the flashlight, the projection would be a hexagon:
So projections are very dependent on the orientation of the object. Four
dimensional object have projections in the three dimensional space. These
projections are dependent on the orientation of the four dimensional object.
Below are projections of a hypercube in three dimensional space (gotten from
www.geom.uiuc.edu).
A third way of visualizing a four dimensional polytope is the idea of a
slice or cut of the object. In the story “Flatland”, the sphere enters the two
dimensional plane world of the square, and all the square sees is concentric
circles of varying size as the sphere passes through the plane. This is the idea
of a cross section or a cut of the object. The same results would be gained by
slicing the object with a knife at various points. (the crossing through the
plane analogy is a little less violent…)
This same idea can be applied to a four dimensional object into the third
dimension. Below is some examples of the slicing of a hypercube into three
dimension, gotten from www.geom.uiuc.edu.
These are just a few ways to try and wrap our minds around what a four
dimensional object would look like. Trying to imagine and perceive these
things is a very difficult thing to do.
This has been a brief introduction into the 4 dimensional world that is
not that far out of our grasps. There are people who have claimed that after a
lifetime of study, they were able to perceive the fourth dimension. There are
objects, like a true Klein Bottle which has only one surface, that are only
creatable with a manipulation of the fourth dimension. The fourth dimension
only exists theoretically for now. But, that’s not to say that it will stay that
way.