The perfect number.

44

Make a loop 44 units around.

Measure the radius.

A perfect Shape for a monument.

But her architects told her they could not build this. She would have to change the design.

So she formed her 44 circumference circle into a square,Each side 11 units.

And these are proportional to the actual measures of the great Pyramid of Egypt. The ratio of the perimeter

of the base to the altitude is exactly 44

7or 2 𝜋.

If each unit on our model was a cubit (the distance from fingertips to elbow),the great pyramid is a 40 times enlargement of our model.

Consider the red triangle.

Here is a model of the great pyramidwith the red triangle.

Check it out. The ratio of two sides of the red triangle is the same as theRatio of two parts of a star (to three decimal places.)

And this is the famous golden ratio. (to three decimal places.)

I’ll ask my doctor.

Make this model of a cone.

EllipseCircle

Parabola

Cut out and assemble this model of a box.

What is it?

What is it?

Imagine each square is a cubit…and each dimension was 10 times larger.

What is it?

Imagine each square is a cubit…and each dimension was 10 times larger.

And how many animals will this ark hold?

Fold this model into a pyramid.Make three of these in different colors.

A pyramid is a third of a box.

A New Figure for Starters

I wish to introduce a new figure, I don’t believe it’s in Math books and it doesn’t have a name. Yet it is the basis of entire fields of mathematics. And we can use it to grow 3-D figures.

You can name the faces like this.

And you have a two sided die.

You can name the faces like this.

And you have a binary random number generator,

The faces are usually called heads and tails.

All probability theory can be generated from this “coin.”

In any polygon, the number of vertices equals the number of edges:

V = E

For a coin, we also have faces. There are 2 of them so we have:

V + F = E + 2

I will make a conjecture that this equation hold for all polyhedrons.

Models can be constructed.I prefer to grow them. We start with something very simple – a seed – and by a series of transformations, turn it into more and more complex figures. The coin will be our seed.

Vertices + Faces = Edges + 2

Coin 4 2 = 4 2

Coin to Pyramid +1 +3 = +4

Pyramid 5 5 = 8 2

Place the coin (heads up and tails down) on a surface. Raise just the heads up to form a box.

Place the coin (heads up and tails down) on a surface. Raise just the heads up to form a box.

Vertices + Faces = Edges + 2

Coin 4 2 = 4 2

Coin to Box +4 +4 = +8

Box 5 5 = 8 2

Next transformation:

Each vertex of top coin will be joined to two vertices of bottom coin.

It is called an anti-prism.

The New World Trade Center is an anti-prism.

Make a model.

Vertices = 8 Faces = 10 Edges = 16

Vertices + Faces = Edges + 2

Are you convinced that the conjecture is true?

Vertices + Faces = Edges + 2

We started our transformations with a coin. You can verify that if instead of a coin, you did these transformations on one face of a large polyhedron, the additions would be the same to our conjecture. For example.

1 face of the old ugly polyhedron vanished, 4 new faces appeared,So we added 3 faces, 1 vertex, and 4 edges.

V + F = E + 2+1 +3 +4

4 was added to both sides.

Does this look like mathematical induction?Can we conclude that our conjecture holds for all polyhedrons?

If our conjecture is true for the old polyhedron, it is true for the new.

1. Cut out and tape this model.

2. Fold it and tape it to form a hollowtriangular tube:

3. Make 2 more of these so you have 3.

4. With 3 pieces of tape, tape themtogether so you have a triangular frame that looks like this:

Do these numbers. satisfy our

conjecture? If not, what went

wrong?

Vertices = _______

Faces = _______

Edges = ________