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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:
http://www.123dapp.com/AssetManager/Publisher?stgAction=getProduct&intProductId=603629

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:
http://www.rhino3d.com/download.htm

...or some excellent, Free 3d-modeling software at:
http://www.123dapp.com/create

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. 

Step 2: Build Some Boxes

1. Open Rhinoceros 4.0

2. On the bottom navigation bar click Snap so the model creation coordinates will snap to the grid. (Figure 1)

3. On the bottom navigation bar click Ortho so the model creation coordinates will be symmetrical. (Figure 1)

4. Click Box tool on the left navigation bar to create a rectilinear solid. (Figure 2)

5. Click on three corners of box. (Figure 3)

6. Right Click on Zoom Extents to zoom into model in each quadrant. (Figure 4)

7. Click on three corners of second box. (Figure 5)

Step 3: Boolean Subtraction

8. Click on second box to select it. Then click on Move tool. (Figure 6)

9. Click on grid, then click second time one unit down to Move box. (Figure 7)

10. Right Click on Boolean tool. Left Click on Boolean Difference. (Figure 8)

11. Left Click on smaller box, then Right Click. (Figure 9)

12. Left Click on Zoom Dynamic, then Right Click. Move cursor over Lower Left Quadrant. Hold down left mouse button and drag down to zoom out. (Figure 10)

Step 4: Copy and Paste

13. Select object by Left Clicking on it. Select Copy Tool. (Figure 11)

14. Click in then on the upper part of the Lower Left Quadrant. Click a second time in the lower part of the quadrant. (Figure 12)

15. Click three corners of another box. (Figure 13)

16. Select Copy Tool. Click to make 3 copies of box in other 3 corners. Add a fourth copy to the side. (Figure 14)

17. Hold the Shift key, then select each of the 6 objects composing the now-created cube.

18. Select Edit > Group > Group.

19. To make a copy of the cube, hold the Control key, then click the C key. Then, while still holding the Control key, click the V key.

20. Click Scale. Click in the center of the cube in the Upper Left Quadrant. (Figure 15)

Step 5: Little Box in Big Box

21. Click to the far right then Click again part of the way to the center of the cube to shrink the cube to a smaller size. (Figure 16)

22. Drag the the smaller cube down to the center of the larger cube in the Lower Left Quadrant. (Figure 17)

23. Select the box off to the left side. In the Upper Left Quadrant, drag the box straight down so that it is centered on the horizontal, red construction plane line. (Figure 18)

24. Transform > Taper. Click near the bottom of the box. Click near the top of the box. Click to the right of the bottom of the box. Move the cursor left to taper the top of the box. (Figure 19)

25. Click Rotate. Rotate 45 degrees to the left in the Lower Left Quadrant, the Front Viewport. (Figure 20)

Step 6: Extend and Taper a Box

26. Click Rotate. Rotate 45 degrees to the left in the Upper Left Quadrant, the Top Viewport. (Figure 21)

27. Move the narrow end of the diagonal box to a corner of the small cube. (Figure 22)

28. Turn off Ortho. Hold down the Scale tool. Select Scale1D. Click in the upper left end of the diagonal box in the Lower Left Quadrant, the Front Viewport. Then Click on the lower right end. Drag the end to a smaller size. (Figure 23)

29. In the Upper Left Quadrant, rotate the diagonal box so it intersects the corner of the big cube. Do the same in the Lower Right Quadrant and again in the Lower Left Quadrant if necessary. (Figure 24)

30. Hold down the Scale tool. Select Scale1D. Click on the upper left end of the diagonal box in the Lower Left Quadrant. Then Click on the lower right end. Drag the end to a smaller size. (Figure 25) Repeat in Lower Right Quadrant.

Step 7: Copy and Paste

31. Rotate as necessary in the Top, Front, and Right Viewports. (Figure 26)

32. Right Click on Zoom Extents to zoom into model in each Viewport. (Figure 27)

33. Click Snap. Click Ortho. Transform > Mirror. Click twice on the center line which vertically bisects the cubes in the Top Viewport. (Figure 28)

34. Hold the shift key and select the newly created diagonal box in the upper left corner of the Top Viewport. (Figure 29)

35. Transform > Mirror. Click twice on the center line which horizontally bisects the cubes in the Top Viewport. (Figure 30)

Step 8: Finish and Save

36. Hold the shift key and add the two newly created bottom diagonal boxes to the selection in the Top Viewport. Edit > Groups > Group. Transform > Mirror. Click twice on the center line which horizontally bisects the cubes in the Front Viewport, a.k.a. Lower Left Quadrant. (Figure 31)

37. Right Click on the Shade tool to shade all viewports. (Figure 32)

38. Double-Click the word Perspective to maximize the Upper Right Quadrant perspective window. Hold down the right button to rotate the objects to check and examine the final product. (Figure 33)

39. Double-Click the word Perspective again. File > Save As > Save as type .STL > Save

Step 9: Print It UP!

40. Upload to www.shapeways.com for 3d printing in material of choice.

Some of the available materials include nylon, alumide, stainless steel, gold-plated stainless steel, bronze-infused stainless steel, sterling silver, ceramic, ABS plastic, ceramic, gypsum composite, and so on.

I ordered mine in Shapeways' "White, Strong & Flexible". It is inexpensive. It is strong and flexible. Here's the description:

http://www.shapeways.com/materials/white_strong_flexible



endnote:
Physicists usually do not speak of "moving" or "traveling" through time, saving those terms for changes in spatial position as the time coordinate is varied. The speak of worldlines that form closed timelike curve loops in spacetime, allowing objects to return to their own past. However, movement in time is interrelated to spatial movement.

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More by TimeJourneyTools:Stop the World & Melt With You TJT6/6 3d Nocturnal Celestial Stardial TJT5/6 3d 4-Dimensional Tesseract Hypercube Model B TJT4/6 
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