A cube is usually represented in the plane as:
(I think this is called a “net”)
Is there a reason why the following representation is not used?
No, no reason at all that I know of; there are several other ways to do it as well.
(I was pondering this question myself a while back, because I was making dodecahedra for the kids.
Sometimes thought, the net of a solid is laid out in a specific way so that several of them can be tesellated efficiently on a large sheet of card and cut out with minimal waste.
The cross-like version perhaps folds up more naturally into a box with four sides and a lid, the base being the centre of the cross. So it might be for ease of construction or to enable you to leave the lid open. But if you glue all the edges up, it doesn’t matter, as Mangetout says.
I like you Mangetout, anyone who makes dodecahedra can’t be all bad.
To the OP: you’re right – the net you show will form a cube, and besides that the cross may have asthetic appeal over the long tee, one other reason springs to mind: If you form the cube from card or paper using each net, you have the same number of joins (of course) but for the long tee the joins have a greater continuous length unsupported by folded edges, and, assuming joined edges are weaker than the folded edges, the overall structure is correspondingly weaker.
Given that I only just formed this conclusion by drawing very sketchy sketches on the back of a cigarette packet, I’d take it all with a pinch of salt.
It started with octahedra; I was sending out some rare seed potatoes to a few friends and didn’t want Her Majesty’s parcel mangling service to destroy them in transit; I settled on a regular tetrahedron, which, although it was only made of corrugated card and sellotape, was tough enough to step on without crushing. from there, dodecahedra seemed the next logical step (the kids were so thrilled with them that they coloured them in and wouldn’t part with them (they even took them up to bed).
Next time I have something vaguely round to send in the post, it’s going in a dodecahedral parcel 
oops, octahedron, not tetrahedron (there wouldn’t be enough space inside a tetrahedron to make it worthwhile)
I don’t know why, but that strikes me as one cool word.
“Tesellated”. I love it!
Thanks for your comments. I guess it’s mostly an arbitrary usage albeit with potential structural stability a la The Great Unwashed’s post.
This arose while I was helping my daughter with her homework. I will not complain here how distressed I was to learn that she was being taught that a cube is NOT a rectangular prism (and that a square was NOT a rectangle).
I used to have a book called Brain Boosters. It was published by Natural History Press (publishing arm of the American Museum of Natural History), and it was filled with science puzzles. One of the more interesting ones involved the many different ways of laying out what you call the “net” – don’t limit yourself to cases where you have four squares in a row – there are lots of other ways.
A related problem is disassembling a tesseract (a four-dimensional analog of a cube) in three dimensions. There are a lot of ways to do this. The classic one looks like your “net” for a cube – you pile up four cubes into a stack, then place four cubes around the second one down in the stack. Salvador Dali showed Christ crucified on a “cross” made of such a disassembled tesseract. The other classic way to do it is to place one cube inside another (they’re distorted to be very different sizes in this dissection), then join the corresponding corners with straight lines. The six “trapezoidal” figures, along with the inner and outer cubes are your constituent cubes.
A (techy) mate of mine used to use it in his chat-up line; “Hey baby, I sleep star shaped, you better learn to tesselate”
KarlGauss :eek: I can’t believe that they would say that; did you get her to point out the meaning of the word ‘regular’?
Actually, I think there is no good way to tesellate either the cross or T versions of the cube, however, there is a good way to tesellate a half-and-half mix:
X
XXXXx
TXxxxx
TTTTxt
TXtttt
XXXXxt
Xxxxx
x
etc.
The T does not tessellate, but the cross most certainly does. Try something like this:
aaaBcddddefgggg
aBBBBhdeeeeigjj
kklBhhhhmeiiiin
llllohmmmmpinnn
The story He Built A Crooked House deals with a house built in the shape of a dissassembled tesseract. Things get interesting shortly after construction is done. The story’s short, well written and worth the read. If only I could remember the author.
Maybe I missed something here. Both the cross and the *t *shapes can tessellate the plane. Were y’all talking ‘bout sumpin’ else?
Robert A. Heinlein, dammit.
You can make polyhedra out of drinking straws and paper clips. Use clips that are almost half the inner circumference of the straws, so that they fit snugly and don’t slip out. You can also bend the clips a little to make them wider. Calculate how many straws (edges) and clips (edges * 2) you need. Now figure out how many edges meet at a vertex, and put that many clips together in a star pattern (think of putting keys on a key ring). Finally, put a clip in the end of a straw, and repeat until you are done. You can tie a thread to a clip and hang the polyhedron with it. The easiest polys to build are tetra-, octa-, and icosahedra, because the triangular faces make a rigid framework. For other polys you either have to make them stellated or work out an internal framework. You can cut straws to shorten them, or put two together by flattening the end of one, folding it in half along the axis, and putting it in the end of the other. Push it in a ways and you’ve got a fairly secure join.
For insights on geometry and ideas for more fun projects I suggest The Dymaxion World Of Buckminster Fuller.
He invented the geodesic dome, tensegrity, and an unbelievably cool van that got 40mpg.
I know about geodesics, but please to explain… .what is tensegrity?
My copy of Dymaxion is on loan at moment so this will not be the greatest of explanations.
Tensegrity- a structure in which rigid parts push out and wires pull in. EG a traditional mast is a single solid beam. Fuller’s tensegrity mast consisted of metal units (imagine an X given a half twist) connected by steel cables. No metal unit touched any other but the mast is sturdy.
I’ll look for some links.