| What polygon do you get if you slice off a single vertex of an icosahedron? Podecahedron pentagram
At the Hotel Cardinality, there are as many rooms as natural numbers and they are all full for tonight. If Ryan wanted to spend the night, show how the manager could make room for him. Ryan could stay with one of the natural numbers because they would correspond according to cardinality. Give him a room with a rational number.
Compare the cardinality of the natural numbers and the rational numbers and show why. The cardinality is the relation between the numbers so for the rational numbers there is a fraction as to where natural numbers just have 1. [1,2,3,4,5,...] [1.1,1.2,1.3,1.4,1.5,...] Rational numbers make up natural numbers. The cardinality will be 1-1 with corresponding pairs. Both have the same cardinality since infinity is taking place.
Consider a 100x100 grid. What rectangle can you make on the grid (using whole number lengths) that is the closest to a Golden Rectangle? Because each rectangle gets bigger as you add on. 161.803 x 100
Show mathematically why this construction gives us a Golden Rectangle. 
 Because of the curve, that is what is adding the extra .618 to the square, thus creating the Golden Rectangle. b/c in a golden rec the sides have to match up so it being 1+1=2 and the other side is 2 knowing this the other two sides will be two The first shape is a reg polygon-a square-by adding the angle the correct amount of length is added to the next side-creating a Golden Rectangle. Because the curve is the mathematical curve and when the rectangle is attached to the square, it makes a golden rectangle. 1/1, 1/2, 2/2=1.64 Because it gives us the lengths and all that match up with that of a golden rectangle. Because there is a golden spiral forming and the proportions look correct, when you take a square out you get the same dimension proportion. You keep adding on to the area you started with and it gets bigger and bigger. 2/2=1 3.3/2=1.65 You need to add on the cube to make it a Golden Rectangle.
Use a tetrahedron to show how Euler's Formula works comparing faces, edges, and vertices.
5-4=3 which equals a triangle. Because the triangle fits inside it evenly and creates a right angle.
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