From the Archives – 31 Ways to Halve a Cube

I’ve had several people ask me, “Your site looks really great, but where’s your art?”  Since all of my current projects are in various stages of “half-baked”, I decided I’d look around at some of my past projects to look for something interesting to share. And thus is born in the first of a new series of posts called “From the Archives”

Plumbing the Depths of the Lowly Cube

One day almost 12 years ago I got to wondering, “I wonder how many ways there are to divide a cube into two identical halves?”  The literal answer is “infinitely many” – for example, just take a straight slice through the center of the cube from any direction whatsoever. So I guess the real question was, “how many interesting / beautiful / surprising ways are there to divide a cube into two identical halves?” This page of sketches shows that I came up with over thirty options in one sitting:

Halved Cubes sketch - Phil Webster, 1990

 

And a few of the more interesting ones I went on to prototype in cardboard – see the pictures below as we go along…

In the course of creating those sketches, you can see that I identified several “families” or “themes” for how to generate ideas.  Let’s take a quick look at each group.

Categories for Halving a Cube

Single Plane

This is the simplest approach: simply slice the cube in half using a single straight cut. Even so, there are at least six different ways to do this that are qualitatively different from each other (1-6 in sketch). Number 7 is a variation on 3 which is no longer a single plane cut, so it doesn’t really belong in this category. Oh well! :-)

8-Cube and Derivatives

If you imagine a cube being cut in half in all three direction you get 8 “sub-cubes”. Combining these in different groups of 4 leads to possibilities 8-11. Numbers 10 and 11 are in parentheses because they only work on paper, because they’re attached only along edges.  In the case of 11, you’d have to pass the two halves through each other to reassemble the cube! Here is Number 12, which also incorporates some diagonal elements:

"Halved Cubes #12" by Phil Webster

Main Diagonal

The “main diagonals” of a cube (each cube has four of them) run from one corner to the corner at the far opposite end of the cube. Number 19 and 20 are built using these diagonals, and number 21 is actually a hybrid with the last category, using the main diagonals of the 8 sub-cubes.

Tetrahedral

Every cube has two tetrahedra “hidden inside” it by connecting two sets of the four diagonals on the faces of the cube. Numbers 22-26 explore halves that use some or all of these tetrahedral edges in their construction. Here is my model of Number 22 (an especially cool one in my opinion!):

"Halved Cubes #22" by Phil Webster

Rectangular Prism

In the same way that you can split a cube into 8 sub-cubes, you can use other combinations of division along each of the three directions to divide the cube into rectangular prisms (a.k.a. boxes!) instead. Number 27 looks at a possibility if you cut the cube in 3 along two sides and in half along the third, creating 18 sub-boxes. Number 28 takes an entirely different approach, imagining boxes or “columns” within the cube connected by slanty surfaces. Number 29 is frankly out of place – it doesn’t seem to have anything to do with prisms at all!!

Donut

What if each half had a “hole” all the way through it, forming a “donut” (or in math terms, a “torus”)? Number 30 is an example of this with the goal of the hole being in the center of the face of the cube.  This model is one of the ones I built and it’s great fun because if you present it to someone assembled:

"Halved Cubes #30" (assembled) by Phil lWebster

and ask them to guess what the halves will look like, most people are surprised by what they find when they pull it apart:

"Halved Cubes #30" by Phil Webster (separation sequence)

Fractal

Number 31 explores the idea of a “fractal” or self-repeating approach. This category is technically unbuildable since the parts become infinitely small, but of course you could build an approximation of the one I sketched given enough time and patience. But I never did. :-)

A Few New Possibilities

That completes the tour of this sketch “from the archives.” But before I sign off, here are a few new possibilities that occurred to me as I looked through these old sketches – mostly extensions of the existing ideas above.

N-cubed Cubes

If you take the 8-Cube idea and divide the original cube into 3, 4, 5… along each direction instead of just 2, you get 27 (3x3x3) sub-cubes, 64 (4x4x4) sub-cubes, 125 (5x5x5) sub-cubes, etc. In fact, on closer inspection, Numbers 13 and 14 are already based on the 4x4x4 case. The odd-numbered ones tend to be a little less elegant because the central sub-cube has to get split in half (can you figure out why?), so here is Number 13 extended to the 6x6x6 case (anyone remember Qbert?!):

Halved Cubes - 6x6x6 example

Zig-Zag Cuts

Instead of the single-plane cut of the first category, we could instead use 3 cuts in a “zig-zag” fashion like this:

Halved Cubes - "zig-zag" example

 

This could be extended even further by adding more zigs and zags to make a “lightning bolt” cut:

Halved Cubes - "lightning bolt" example

Curved Cuts

And finally… every example above limits itself to straight edges and faces. But why not use curves?:

Halved Cubes - "curved cut" example

 

Whoa, that really opens up the possibilities…  Can you think of any others? If so, leave them in the comments below!

 

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About Phil Webster

Phil is the creator of GeometricArts.com. You can reach him on the Contact page.
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