Gee, I'm a Tree! Trivia Quiz

Answer: Rectangle

Another way of defining a rectangle is a shape where the diagonals both bisect each other and are equal. While all squares are rectangles, not all rectangles are squares.

Answer: Icosahedron

An icosahedron is a shape with 20 triangular faces, 30 edges (lines where two faces meet) and 12 vertices. The icosahedron, dodecahedron, tetrahedron, octahedron, and cube are the only 5 regular polyhedra. Plato considered these the most perfect of shapes, so they are actually sometimes just referred to as the Platonic solids.

Answer: Mobius Strip

This shape has many interesting properties. First of all, it only has one side! If you started at a point and traced a pencil down the center of the strip until you came back where you started from, it would have actually transversed both "sides" of the strip. Furthermore, if you take a pair of scissors and cut down the center of the strip, you will be left with one piece even after supposedly cutting it in half.

Answer: Klein Bottle

In many ways, this shape is just a 3 dimensional version of the Möbius strip. Just like the Möbius strip has only one "side", the Klein Bottle has only one "side"...it is a bottle which doesn't really have an interior! Don't try making one of these things at home though. Just like a Moebius strip (which around any point is really just a plane figure) can only be made by going into a third dimension, if you wanted to make a Klein Bottle you would have to go into a fourth dimension, which can be rather hard.

Answer: Circle

This is known as the Isoperimetric inequality, and is actually exceptionally difficult to prove. In fact, for many years mathematicians had an incorrect proof! They had managed to "prove" it, but along the way they had to assume that such a shape actually existed in the first place. (This didn't have to be the case...try thinking of a smallest number bigger than 0, for example). Karl Weierstrass came up with the first correct proof in the 19th century, over 2000 years after the problem was first proposed. This is a problem that has practical applications.

For example, imagine a rancher with 200 ft of fence who wants to give his cattle the maximum grazing area possible.

Answer: Torus

For all this complicated description, a torus can perhaps be visualized better as that doughnut you eat every morning (or the unfilled ones, at least).

Answer: Right Triangle

There are a couple of ways of seeing this. Perhaps the easiest (if you know trigonometry) is to remember the formula area=0.5*a*b*sine(C), where C is the angle between sides a and b. a and b are given, so we want to maximize sine C, which occurs when C=90.
Another way to visualize this is holding one given side of the triangle as the base, and rotating the other side. The third vertex will be farthest from the opposite side when the other side is going straight out, i.e. at a right angle.

Answer: Rhombus

Another definition would be a quadrilateral, all of whose sides are equal. A square is just a special case of a rhombus when all four angles are equal.

Answer: Pentagons and Hexagons

One way to see this is to picture what happens at a vertex of the pentagon. We've got at least one pentagon and some number of hexagons meeting. The pentagons have angles of 108 degrees each and the hexagons 120. Unfortunately, there's no way to get a total sum of 360 using these numbers. One thing we could try, however, would be to take a pentagon and two hexagons (total angle=348 degrees) and bend the shape slightly downwards to accommodate the missing 12 degrees. If we did this, we would get a closed shape which mathematicians call a truncated icosahedron, chemists call a fullerene structure, and normal people just call a soccer ball.

Answer: Icosapentahedron

The other three shapes are what are termed Archimedean Solids (like the Platonic Solids, but with two or three types of faces instead of one).
The rhombitruncated icosidodecahedron has dodecahedrons, squares, and hexagons. The snub cube has squares and triangles, while the pentagonal hexecontrahedron has 60 slightly deformed pentagons.

I hope you had fun with this quiz!