Me and my friend were just discussing usual kind of things and somehow this came up in conversation. It seemed a simple enough question to ask ourselves but we found ourselves struggling to come up with any regular 3-dimensional shape with only 3 sides. The least we could imagine was four, a simple triangular based pyramid. So we decided to leave it there and I made a mental note to look it up on the internet when I got a chance.

Now i did a very small bit of research before trying to work it out, well actually I say research, i just found a forum online and had a read through, there seemed to be quite a few experts, or knowledgeable people there, which knew a thing or two about geometry and that sort of thing.

Lets try and answer this question in a simpler way first, can you have a regular 2-sided, 2-dimensional shape. So in this question we are looking for a regular 2-dimensional shape, by regular i mean like a square, a pentagon, an octagon. The fewest sides i can imagine a 2-dimensional shape having would be 3, a simple triangle. But you can say you can get a semicircle, where the curved edge counts as 1 side, and the flat edge as the second side. In which case you are then using a 1 dimensional line and a 2 dimensional curve to get your 2 sided object. So we conclude that you can only get a 2-sided, 2-dimensional shape using atleast one 2-dimensional curved line. And by only using 1-dimensional lines we can obtain only a three-sided shape as the least sided shape. Follow that? I’ve pretty lost managed to lose it myself just thinking about how to put it.

So, ignoring curved lines, the minimum sides for a 2-dimensional shape is made of 1-dimensional lines. Now if we think about a simple regular 3-dimensional shape, like say a cube, it is made of 2-dimensional squares. So we can guess that a regular 3-dimensional shape, ignoring curved surfaces, is only made of 2-dimensional surfaces. As a little added extra bit i’m just thinking does that mean that to get to a 4-dimensional shape we would need to make it out of these simple regular 3-dimensional shapes, ignoring curves? Hmmm that is an interesting thought. But impossible for me to imagine, and most humans i would hazard a guess.

So anyway, from there i tried to imagine a regular three-dimensional shape made of triangles…..I couldn’t……I retreated back to the forum. There i found someone had posted something about seeing a 3-dimensional object from one angle you could see 3 sides had to all interact at one point, or something like that i forget exactly what it was. But it got me thinking, when you look at a cube directly onto a corner you see 3 faces which all meet at that corner, so what if you could bend them all round again to interact at a corner on the other side? I couldn’t imagine it with a cube, but i quickly realized i could just think of a rugby ball, except with it having 3 curved surfaces instead of 4. I’d done it! I’d managed to imagine a 3-sided 3-dimensional object. Although i did have to bend the rules a bit, haha.

I think i’ve come to the conclusion that:

- I can’t do research very well at all, and
- There is no 3-sided, 3-dimensional object which only uses 2-dimensional shapes.

I don’t like the use of curved surfaces, because it just feels like cheating, once I could break that invisible rule i kind of set for myself i found i could imagine many things, for example: a two-sided 3-dimensional shape would just be 2 intersecting spheres, and you could either use the non intersecting or intersecting parts to get your shape. And another simple 3-sided, 3-dimensional shape could be a cylinder, but it doesn’t look as nice. There you go, take what you will from that.

Here‘s a link to that forum i so reference. They start talking about hollow 3D shapes aswell, which i just completely ignored for my own purposes.

For a 3 dimensional shape that has only 3 faces, try this:

(A) roll up a piece of paper into a fairly tight tube or cylinder

(B) bend and crease the midway point along its length…so that it Looks like a 10 in. straw that has been folded in half — giving it a 5 inch total length.

(C) Now… work your way out from the center and repeat the crimping/folding EXCEPT…. on the 2nd bend, make it 90 degrees different than the first bend.

This creates a chain of little 3d shapes.– each made up of three triangular faces.

Imagine a long, hexagonal cross-sectional piece. Twist it by 180 degrees. Join the ends together.

Lo. A 3 sided, 3D shape.

Do it with a square for a 2 sided one.

Essentially take a Möbius strip into 3d.

A cylinder?

thats what i was thinking

I can do you one better a two sided three dimensional object… a mobius strip!

I can do even better than that with a one sided 3 dimensional shape. a klein bottle

i would go for a prismatic toroid. also called a yahrun :)

Technically, that would be a 4 dimensional shape, the 3d “klein bottles” are just 3 dimensional representations of an actual klein bottle.

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