User:AWrinkleInTime

How do you create a tesseract? The Schlegel Diagram and the Wikipedia Tesseract page have a number of issues which the Warner Model (see image) addresses.

There are four dimensions represented: X, Y, Z, and K. There are eight cubes wrapping around or bounding an inner four-dimensional void. The cubes are defined by the sixteen vertices from [0,0,0,0] to [1,1,1,1]. Note that only one of the eight cubes would exist in our 3-brane. The other seven have a K component, making them a part of a different 3-brane. The black inner cube is a symbol representing the four-dimensional void being bounded.

Assume the lower purple cube exists in the [X,Y,Z] brane at K=0 and is in "our" 3-brane. It has the vertices from [0,0,0,0] to [1,1,1,0]. Then the upper purple cube exists one unit distance away and is its opposite, located opposite of the bounded four-dimensional void. In other words, its points are located from [0,0,0,1] to [1,1,1,1].

The lower purple cube's right face is in the y-z plane. Two-dimensional faces bound a 3D cube and "hinge" on one-dimensional lines, turning ninety degrees into a different 2-brane. Similarly, three-dimensional cubes bound a 4D octachoron and "hinge" on two-dimensional faces, turning ninety degrees into a different 3-brane. You can follow one of the blue ellipsoid rings from the right face to the adjoining face of a different cube in a different 3-brane.

In this example, the adjoining cube is a red cube in the [X,Y,K] 3-brane. The right side of the purple cube is the bottom side of the red cube. The red cube is not in our 3-brane and would not be perceptible to our senses. Traveling "up" in the red cube, one moves from K=0 to K=1, and enters the right side of the upper purple [X,Y,Z] cube at K=1. A person standing upright and facing to the right in the lower purple cube would emerge upside down and facing the opposite direction (left) in the upper cube.

The blue ellipsoid rings then show the six ways that one can traverse around the perimeter of an octachoron. A 3D cube has three ways for circling around it; an octachoron has six.

There are no "inside" or "outside" faces on the bounding cubes of an octachoron. All eight cubes fit together so that each of the forty-eight faces mate with an adjacent cube's face. There are therefore twenty-four distinct match-ups or shared faces. The sixteen vertices define eight separate cubes. In this example, only the two purple cubes have distinct vertices, and all the other cubes are defined using shared vertices. It is always the case that a given cube and its opposite have distinct vertices, and the other cubes are then defined using shared vertices.

This particular model is printable on a 3D printer. The red tube connecting the two gold cubes (in K Dimension) through the black cube is a structural support. It is not a part of the mathematical model, but rather is used to help hold the gold cubes in position, in a physical model.

Inside and outside is always defined in the fourth dimension. For the lower purple [X,Y,Z] cube, "outside" would be any real number value less than K=0. "Inside" would be real numbers between K=0 and K=1. For the upper purple cube, "outside" would be any real number greater than K=1. For the red [X,Y,K] cube, inside and outside are similarly determined but relative to the Z Dimension.

Regarding the Shlegel Diagram, it is arguably mathematically correct but based on hidden assumptions, or else is mathematically incorrect. Either way, it is misleading for a variety of reasons. For one, there are illusions created by hidden lines.

Imagine looking at an orthogonal drawing of a cube edge-on. It looks like a square. The square at the center of the Schlegel Diagram is actually the two gold K Dimension cubes—as projected orthogonally edge-on in the fourth dimension. It gives the misleading perception that there is only one cube, and that cube is located at the center of the octachoron. In actuality, there are two cubes bounding a four-dimensional void (gold cubes in Warner Model). In effect, that model falsely insinuates that there is no central 4D void.

The four perimeter cubes thus are spatially located between the two gold cubes in a "belly ring". The near and far cubes in actuality round it out to tally to eight cubes (again concealed by hidden lines but also bounded by the outer perimeter lines of the perimeter cubes). Note that the near and far cubes share the same axis (for the purpose of the visualization) as the two central gold cubes. In this way, one spatial dimension is forced to do double-duty; this is a type of distortion introduced by the act of projecting a four-dimensional object into two dimensions.

The diagram suggests only seven cubes, with the central cube being smaller than the other six. It insinuates that the composition of the seven together forms the eighth—the totality is the eighth. This is misleading. All the cubes are the same size and none are the composition of the others.

Having said all that, one can say that actually there are eight 3D cubes, one 4D central void, and then a composite 4D object that is the collection of the eight cubes plus the central void--the octachoron proper: (8) 3D cubes + (1) 4D void = (1) 4D object.

Likewise, the Schlegel Diagram insinuates that there are "inside" and "outside" faces. This again is erroneous due to looking at an orthogonal drawing edge-wise. Without exploding the diagram apart and rotating it, these look like end-point vertices. Were it exploded apart, one would see the distortions of that projection from 4D space to 2D space. The "inner" face of a peripheral cube adjoins the face of a gold cube that points in its same direction. It's "outer" face adjoins the equivalent face in the other gold cube. As such, the peripheral cubes are additionally distorted akin to a u-shaped extrusion.

The "outer vertices" of a peripheral cube would then be shown to be tangent points on arcs from a given peripheral cube bending 180 degrees from one gold cube to the other in the fourth dimension. Note separately that the near and far faces of a peripheral cube mate with their adjacent faces on the near and far cubes. On any of all six peripheral cubes, it is the "inner" and "outer" faces that mate with the pair of gold (central) cubes.

We also see distortion in the Schlegel Diagram by the forty-five degree angled sides of the peripheral cubes. The "outer" faces of the peripheral cubes must make a 180 degree turn, since the K Dimension is composited on top of another dimension in our 3-brane for the purposes of our projections. In other words, there is mathematically a ninety-degree turn from one 3-brane to another, but in the diagram the K Dimension makes an additional ninety-degree turn so that one of the dimensions performs double duty as also representing the K Dimension. The only way to do that is with a U-shaped arc.

The Warner Model mitigates this issue by exploding the octachoron apart and representing the K Dimension as offset forty-five degrees from each axis, with the K axis effectively rotated before projection. (In this way the "double duty" is split equally amongst all three dimensions and the hidden lines illusion is avoided.) Thus, only a 135 degree turn is needed, and this is able to be shown with the blue ellipsoids in a way that is consistent and uniform for all axes.

What would it be like to live inside an octachoron house? Check out my books or contact me for more details.

Official Wikipedia Page: Tesseract