This instructable explains how to produce a 3d model of a 4d cube. It can also be described as a 3d shadow of a 4d model. It is the second of two 3d models of 4d objects I'm uploading. It is *Time-Journey Tool 4 of 6.*

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

http://www.123dapp.com/AssetManager/Publisher?stgAction=getProduct&intProductId=603631

If you wish to 3d-model your own 4d hypercube, I can offer instructions for 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:**

ï· Computer with internet access

ï· 3d modeling software (or download my design for free)

ï· access to 3d printer (I use Shapeways.com)

We experience our lives in 3 dimensions. Einstein showed we live in 3 spatial dimensions with a 4th dimension of time. This is the spacetime continuum. The 4th dimension upon which this instructable instructs is the spatial 4th dimension.

Dana Carvey as "the Church Lady" used to ask something like, "isn't that spatial?" Indeed it is.

Were we stick figures living out our lives, strolling around on a 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, back on the sheet-of-paper Universe.

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

http://www.123dapp.com/AssetManager/Publisher?stgAction=getProduct&intProductId=603631

If you wish to 3d-model your own 4d hypercube, I can offer instructions for 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

ï· Computer with internet access

ï· 3d modeling software (or download my design for free)

ï· access to 3d printer (I use Shapeways.com)

We experience our lives in 3 dimensions. Einstein showed we live in 3 spatial dimensions with a 4th dimension of time. This is the spacetime continuum. The 4th dimension upon which this instructable instructs is the spatial 4th dimension.

Dana Carvey as "the Church Lady" used to ask something like, "isn't that spatial?" Indeed it is.

Were we stick figures living out our lives, strolling around on a 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, back on the sheet-of-paper Universe.

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

0 – A point is a hypercube of dimension zero.

1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.

2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square.

3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.

4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).

0 – A point is a hypercube of dimension zero.

1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.

2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square.

3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.

4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).

any ever tried to make a 3 dimensional shadow of a 4 dimensional shadow of a 5d cube?

Wait... Isn't it impossible to create a hypercube in the 3rd dimension? Anyways, Nice Instructable!

Hey Meta, yes it is impossible. But that's what's so cool, it is <em>not</em> impossible to have a <em>shadow</em> of a 4-dimensional object in 3-dimensional space. Here how it works:<br> <br> a 3-dimensional object (a cube) can cast a 2-dimensional shadow (a square)<br> <br> a 2-dimensional object (a square) can cast a 1-dimensional shadow (a line)<br> <br> a 1-dimensional object (a line) can cast a 0-dimensional shadow (a point)

First of all, this is freaking awesome. <br>Second, I really really really want to make this, but i dont have a 3-d printer. So i am planning to just assemble it out of something (probably wood) so i need a definite shape. So correct me it i am wrong, but the basic structure is just two cubes that are linked together from one corner, with lines connecting their vertices, right?

Chime--correct. <br> <br>Once built, take it outside on a sunny day (you guys all still do that, right? Go outside when the sun's out?). The shadows this thing casts are also amazing!!

Lol yes we go outside. Thanks for the advice, i probably never would have thought about looking at the shadows

The shadows of the hypercube that's "Shifted up and off to the side" are much better then the cube-in-a-cube version (they're actually pretty boring for that one) <br> <br>It's the collimated aspect of the light that does it. You could also use laser light, though you'd need something to spread it out. The advantage there, of course, is that it's LASER LIGHT: <br> <br>https://www.instructables.com/id/sharks-with-lasers-on-their-frikkin-foreheads

So... instead of having them linked, I should try doing it up and off to the side. So now there is going to be a little square where the Bottom left of one cube, and the upper right of another cube meet, right? Imma try both. Gonna buy a bunch a sqaure dowels or a bunch of pencils or something. Thanks for your suggestion. <br> <br> <br>Also you link did not work. Instructables said page not found. I also could not find the thing in a search.

Yes, but you can just have shapeways.com print it for you. I just went there and made it order-able: <br> <br>http://www.shapeways.com/model/504228/tesseract-fat-fixed.html/?material=6 <br> <br>but sure, you could make your own.

TJT, <br> <br>Did you post the lateral offset one there? It's so much better than the "cube-in-a-cube" version!

this one is a lot more interesting looking, but I guess they are both accepted interpretation/3d shadows of 4d cubes. I made them both to be comprehensive.

Great tutorial. <br> <br>In my opinion Shapeways quality is terrible thou they are cheaper then others so my money goes elsewhere. <br> <br>There are high quality 3d printing services such as http://ponoko.com for USA and http://i.materialise.com for Europe <br> <br> <br> <br> <br> <br>

this is interesting about their quality. They are the only place from which I've ever gotten 3d prints.

Is there somewhere else you could put your model? I was unable to download it

These are pretty small files. I can email them to you. You can email me at robert(AatT]robertmihaly{dawt]com

Thanks! I sent you an email

This looks like a really cool object which I would like to try and print on my RepRap. <br> <br> I tried the link <br> <br>http://www.123dapp.com/AssetManager/Publisher?stgAction=getProduct&intProductId=603631 <br> <br>but sadly I get the message "You do not have permission to perform that action." <br> <br>Is there any other way I can have access to the STL file ?

hi John, <br> <br>I am happy to email both files to you. They are very small. email me at robert(aht}]robertmihaly({[dawt]})com

I love tesseracts! this is way cool... thx :)

@TimeJourneyTools; Hi! 50 Gold Stars for spreading awareness of 4D! I mean, 3D has been around forEVer *coff* <br> <br>Another 50 stars for modeling the hypercube "shadow" in 3D reality. I did one in 3DWeb and took pics of it at the link below. I suppose I could make a video of it rotating but *sigh* you've got that covered too and even so it would still be 4D in 3D in 2D. <br> <br>Seriously; Good on you! <br> <br>Cheers! : ) <br>Site <br> <br>http://sitearm.wordpress.com/2011/12/13/hypercube/

As if the 4d wasn't cool enough, and the observation of 4D in 3D in 2D (unless the 2d is a sort-of 3d [2 spatial dimensions + time dimension])... then you needed to throw in the Organic Rapeseed Oil on Ginger Chicken Recipe... <br> <br>mmmm. I think I'm starting to get too much blood in my caffeine system, since as hungry as you're making me I'm still getting a little tired... I might conk out for sleep deprivation tonight, but I need to look into that ginger chicken action tomorrow

I'd do this, but I don't want to fall into a rip in the space-time continuum. <br> <br>(Actually, I'm waiting for my 3D printer to arrive. Then I'll make it.)

must be nice.

Very cool! Beats the one I made out of balsa-wood stringers back in...about 1976!