This instructable explains how to produce a 3d model of a 4d cube. The model can be described as a 3d shadow of a 4d object. 4d cubes are also referred to as hypercubes or tesseracts.

This is the first of two instructables I'm putting up to show how to make each of two commonly-accepted 3d projections of 4d cubes. For at least a century this has been an accepted understanding of 4d geometry. This instructable is Time-Journey Tool 3 of 6.

If you do not wish to create your own model in 3d-modeling software, you can take the easy way out and download my model for free at:

If you wish to 3d-model your own 4d hypercube, this instructable provides instructions for modeling in Rhino. If you want to try some 3d-modeling software for free, either get Rhino's evaluation copy at:

...or some excellent, Free 3d-modeling software at:

Recommended Materials and Processes:

 Computer with internet access
 3d modeling software (or download my design for free)
 access to 3d printer (I use Shapeways.com)

You can build the model in Rhinoceros 3d software according to the following instructions, or else you can simply download the model I created. It's free. The best part is you can have a 3d printer service print it if you don't have your own printer yet. I use Shapeways.com. (Though, alas, I dream of my own printer.)

A way to visualize the 4th dimension is to consider relationships between dimensions. For example. A line can cast a shadow that looks like a point. A square can cast a shadow that looks like a line. A 3d cube can cast a shadow that looks like a square. So too a 4d hypercube can cast a shadow that looks like the 3d object of this instructable.

Step 1: More Description of the 4th Dimension

We experience our lives with in 3 dimensions. Einstein suggested we live in 3 spatial dimensions with a 4th dimension of time. This is the spacetime continuum. This concept of the 4th dimension suggests it is a different sort of phenomenon than the spatial dimensions.

4 dimensional space is a concept derived by generalizing the rules of three-dimensional space. Relativity of simultaneity is known as eternalism or four-dimensionalism. Eternalism suggests time is just another dimension, that future events are "already there" and that there is is no objective flow of time.

This step of the instructable is included in an attempt to make basic 4-dimensional geometry more clear. If you already understand 4d or don't wish to read it right now, skip to the next step.

It is probably easiest to contemplate the 4th dimension by first considering the lower dimensions:

a.) A point, a vertex, (it could be called a 0-dimensional "cube"). If you "extrude" it, it stretches out into a line segment.

b.) A line segment, with 2 vertices, (it could be called a  1-dimensional "cube"). If you "extrude it, it stretches out into a 2-dimensional square.

c.) A square, has 4 vertices, (it could be called a  2-dimensional "cube"). If you extrude it, it stretches out to be a 3-dimensional cube.

d.) A 3-dimensional "cube", with 8 vertices, is what we think of as a "cube". If you extrude the plane of each of it 6 sides you get sort of a cube with six cubes attached. But the trick is that the outermost plane on each of the 6 cubes form a 8th cube. A 4d cube has 16 vertices.

It is hard for us to visualize a 4-dimensional "cube", also called a tesseract or hypercube. If we were stick figures living out our lives strolling around on our sheet-of-paper Universe, it would be maddeningly confusing for someone to elucidate the "3rd dimension" to us when our entire culture has been defined by "up" & "down", "left" & "right".

Nonetheless, we can do it. Should we wish to transcend the limitations of these measly 3 dimensions we must consider that which we have not previously experienced, on this sheet-of-paper Universe. 

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