Rotating 2D shapes in 3D (video) | Khan Academy
having the same size and shape cube a polyhedron with six congruent, a three -dimensional figure with two parallel, congruent bases connected by rectangles. We have shown in an earlier note that any two dimensional polygon of N Note that the base area of such a tetrahedron is given by |V1xV2]/2. When this area is multiplied by 1/3 of the height related to the third vector one finds the volume of. Regular polygons, by definition, are two-dimensional shapes bounded by built the Four Elements: sharp-pointed tetrahedra give the sting of Fire, Related Based on that foundation, and a few clues from experience, the.
These can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element in this partial order when the vertex or edge is part of the edge or face.
Additionally, one may include a special bottom element of this partial order representing the empty set and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart that is, between each face and the bottom element, and between the top element and each vertex have the same structure as the abstract representation of a polygon, then these partially ordered sets carry exactly the same information as a topological polyhedron.
However, these requirements are often relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment.
Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is also possible to use abstract polyhedra as the basis of a definition of geometric polyhedra.
A realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron.
However, without additional restrictions, this definition allows degenerate or unfaithful polyhedra for instance, by mapping all vertices to a single point and the question of how to constrain realizations to avoid these degeneracies has not been settled.
The only polyhedra for which it doesn't work are those that have holes running through them like the one shown in the figure below. This polyhedron has a hole running through it. Euler's formula does not hold in this case. These polyhedra are called non-simple, in contrast to the ones that don't have holes, which are called simple.
Rotating 2D shapes in 3D
Non-simple polyhedra might not be the first to spring to mind, but there are many of them out there, and we can't get away from the fact that Euler's Formula doesn't work for any of them.
However, even this awkward fact has become part of a whole new theory about space and shape. The power of Euler's formula Whenever mathematicians hit on an invariant feature, a property that is true for a whole class of objects, they know that they're onto something good.
They use it to investigate what properties an individual object can have and to identify properties that all of them must have. Euler's formula can tell us, for example, that there is no simple polyhedron with exactly seven edges.
You don't have to sit down with cardboard, scissors and glue to find this out — the formula is all you need. The argument showing that there is no seven-edged polyhedron is quite simple, so have a look at it if you're interested. Using Euler's formula in a similar way we can discover that there is no simple polyhedron with ten faces and seventeen vertices. The prism shown below, which has an octagon as its base, does have ten faces, but the number of vertices here is sixteen.
The pyramid, which has a 9-sided base, also has ten faces, but has ten vertices. But Euler's formula tells us that no simple polyhedron has exactly ten faces and seventeen vertices.
Both these polyhedra have ten faces, but neither has seventeen vertices. It's considerations like these that lead us to what's probably the most beautiful discovery of all. It involves the Platonic Solids, a well-known class of polyhedra named after the ancient Greek philosopher Platoin whose writings they first appeared.
From left to right we have the tetrahedon with four faces, the cube with six faces, the octahedron with eight faces, the dodecahedron with twelve faces, and the icosahedron with twenty faces. Although their symmetric elegance is immediately apparent when you look at the examples above, it's not actually that easy to pin it down in words. It turns out that it is described by two features. The first is that Platonic solids have no spikes or dips in them, so their shape is nice and rounded.
In other words, this means that whenever you choose two points in a Platonic solid and draw a straight line between them, this piece of straight line will be completely contained within the solid — a Platonic solid is what is called convex.
The second feature, called regularity, is that all the solid's faces are regular polygons with exactly the same number of sides, and that the same number of edges come out of each vertex of the solid. The cube is regular, since all its faces are squares and exactly three edges come out of each vertex.
You can verify for yourself that the tetrahedron, the octahedron, the icosahedron and the dodecahedron are also regular. Now, you might wonder how many different Platonic Solids there are. Ever since the discovery of the cube and tetrahedron, mathematicians were so attracted by the elegance and symmetry of the Platonic Solids that they searched for more, and attempted to list all of them.
This is where Euler's formula comes in.
You can use it to find all the possibilities for the numbers of faces, edges and vertices of a regular polyhedron. What you will discover is that there are in fact only five different regular convex polyhedra! This is very surprising; after all, there is no limit to the number of different regular polygons, so why should we expect a limit here? The five Platonic Solids are the tetrahedron, the cube, the octahedron, the icosahedron and the dodecahedron shown above.
Well to think about it in three dimensions, what I'm going to do is try to look at this thing in three dimensions. So let me draw this same line but I'm gonna draw it at an angle so we can visualize the whole thing in three dimensions. So imagine if this was sitting on the ground.
So that's our magenta line, and then I can draw my triangle. So my triangle would look something like this. So it would look like this. So once again this is five units, this is three units, this a right triangle. I'm gonna rotate it around the line, so what's it gonna look like? Well this and this right over here is gonna rotate around and it's gonna form a circle with a radius of three, right? So it's gonna form, so it intersects, if that was on the ground it's gonna be three again.
Faces, Edges, and Vertices of Solids ( Read ) | Geometry | CK Foundation
And let me draw it down so it's gonna keep going down. We don't want to press the wrong button. So it's gonna look something like this. That's what the base is gonna look like. But then this end right over here is just gonna stay at a point because this is right on that magenta line. So it's gonna stay at a point.