It’s both a triangular pyramid and a tetrahedron, since those are two different ways of looking at the same shape.
It’s a notation used for games which use dice as a random number generator: one die with four faces.
I used to see these a lot for milk about 20 years ago. less popular now. Packed fairly well in larger boxes. It’s called a “tetrapak”. I’m not sure why they picked that name. Maybe it has geometric allusions… Greek word for four in there, for some reason.
In terms of the object being a perfect tetrahedron, the answer is that, in the limit of the materials, it can be. A perfect cylinder made of flat material with a circumference exactly twice that of its length * sin(60) can be folded to form a perfect tetrahedron (if we ignore the question of how the seams are made.)
The folding requires no stretching of the material. Just precise folds. So, in the real world, you just need to make one large enough that the seams don’t matter in the scale of the object, and fold the edges hard.
Oops, that should be cos(60).
If the circumference of the cylinder is 1.0 the length should be sqrt(3).
Isoceles Tetrahedron is what Wolfram calls it.
Because the design is a tetrahedron, and it’s a trademark. Although they’re more well-known probably for the Tetra Brik, which isnot a tatrahedron but is a design icon.
Is a solid with three plane faces even possible? I can’t visualise any working case (assuming ordinary Euclidean 3 dimensional space)
No, the minimum is 4. Just like the minimum for a 2D shape with straight sides is 3.
OK - four faces, just like the picture. So a tetrahedron then. I guess you have to look at it the right way.
Thanks folks
BTW, Euler’s polyhedral formula would tell you your count was off. That says:
v-e+f = 2
where v, e, f are vertices, edges and faces, and works for all convex polyhedra, as well as some non-convex ones. Convex polyhedra are ones whose surface does not intersect itself and any line drawn between points on the surface stay inside the thing. If such a thing has 6 edges and 4 vertices, it has to have 4 faces. Informally, it will actually work for any polyhedron that you can imagine placing a balloon inside, and inflating the balloon so that all of the edges are now drawn on the surface of the balloon.
BTW, that formula provides a nice simple way to show that there are only 5 regular solids, without a lot of tedious mucking about with the possible angles and so on.
Provided that you define “regular solid” sufficiently rigorously. Euclid, for one, didn’t*, and by his definition, there are actually at least seven (the triangular and pentagonal bipyramids meet Euclid’s definition), or more if you allow non-convex solids, or self-intersecting solids.
*Remark on The Elements, book 13, proposition 18:
Whats happening is as the OP says, its a cylinder with pinched ends.
Its the pathologic case of where the ends are so close together, its all end and no cylinder.
The material was curved before being pinched and so what happens is that different numbers of faces lose the curvature and become flatter due to tension on them. This means that other faces gain curvature, and the curvature is shared at the edge so the edge becomes rounded and there’s no distinct change in curvature at the median location.
If you like, that “face” is a splayed cone.
What people are saying is that in common task of making such a shape, it takes some effort to design the tetrahedron’s four triangles and then join them together, and so this is seen in mathematics and so on. People making air conditioning ducts might call such a shape a manifold, or a tetrahedron, as they use a manifold production technique to make it. (taking flat pieces and making the joins of triangles and trapeziums etc that approximate the curved surface desired. )
But as the OP notes these tetrahedron are made by taking a cylinder of plastic and pinching the ends. So where it looks like it has three sides, then its a cylinder with two triangles added by pinching.
But In other specimens the cone is separated into two by an edge and the four triangles of the tetrahedron appear.
In practice, starting with a cylinder is not a good way to make a tetrahedron. Its only happening with these packages because its a flexible plastic AND the ends are close together relative to the cylinders diameter. For that reason, you may just call it pinched ends short cylinder.
However to reverse my previous point. If you take a regular isosceles tetrahedron, and you slit two opposite edges, the surface will open out into a perfect cylindrical tube. No stretching of the material needed. It is quite fun to see it happen.
There’s four regular concave polyhedra (star polyhedra) along with the five regular convex polyhedra
Using Euclid’s definition, and allowing non-convex polyhedra, I can think of at least one infinite family of them.
Bah.
Oh, and so that you don’t get the wrong impression, that’s a great contribution.
But somebody had to say “bah.”
SOME have already said essentially that: :smack:
So, how do the rest of you give a geometrical definition of a plate of spaghetti? :dubious:
Spaghetti …
Serious … they call them “spaghetti diagrams” 'cause all the lines on them look like a plate of spaghetti … like this one of the US strategy in Afghanistan …
