1. Keep an open mind to all possibilities.
Section 4.1 Pythagoras and His Hypotenuse How a puzzle leads to a proof of one of the gems of mathematics.
Keep an open mind to all possibilities.

2. Question of the Day
A baseball diamond is really a square measuring 90 feet on a side. How far does a catcher have to throw the ball to get it from home plate to second base?

3. Pythagorean Theorem
In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

4. The Puzzle Proof
A proof of the Pythagorean Theorem.

5. Often in life, hard questions are made up of many each pieces.
Section 4.2 A View of an Art Gallery Using Computational Geometry to Place Security Cameras in Museums.
Often in life, hard questions are made up of many each pieces.

6. Question of the Day
Here’s a floor plan for an art gallery. If you stand in the corner marked A, which walls or parts of walls can you see? A

7. Question of the Day
Here’s a floor plan for an art gallery. If you stand in the corner marked B, which walls or parts of walls can you see?

8. The Klee Art Gallery Question
For an arbitrary polygonal closed curve in the plane with v vertices, how many vertices are required such that every point in the interior of the curve is visible (directly in the line of sight) from at least one of these chosen vertices?

9. Modified Question:
For an arbitrary gallery having v corners, how many cameras are needed to ensure that every point in the gallery is seen by some camera?

10. The Art Gallery Theorem
Suppose we have a polygonal closed curve in the plane with v vertices. Then there are v/3 vertices from which it is possible to view every point on the interior of the curve. If v/3 is not an inte3ger, then the number of vertices we need is the biggest integer less than v/3.

11. Section 4.3 The Sexiest Rectangle Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle.
Often in life when faced with a difficulty, it is valuable to look for something else that is comparable, but easier to resolve.

12. Question of the Day
Imagine the ideal shape that comes to mind when you hear the word “rectangle.” Draw the shape on the board and initial your rectangle.

13. The Golden Rectangle
What is a Golden Rectangle?

14. The Golden Rectangle
Where can Golden Rectangles be found?

15. The Golden Rectangle Within a Golden Rectangle.
If a Golden Rectangle is divided into a square and a smaller rectangle, then the small rectangle is another Golden Rectangle.

16. Golden Rectangle
Make your own Golden Rectangle.

17. Section 4.4 Soothing Symmetry and Spinning Pinwheels Can a Floor Be Tiled Without Any Repeating Pattern?
Specifying the meaning of a familiar term
or notion can open our eyes to new
possibilities and ideas.

18. What is the most symmetric shape you can think of?
Question of the Day
What is the most symmetric shape you can think of?

19. What is symmetry?

20. What is Rigid Symmetry?

21. What is symmetry of scale?

22. Uniqueness of Scaling.
There is only one way to group the Pinwheel Triangle into super-tiles to create a Pinwheel sper pattern in the plane?

23. No Rigid Symmetry
The Pinwheel Pattern has no rigid symmetries.

24. Coincidences are flashing lights alerting us to potential insights.
Section 4.5 The Platonic Solids Turn Amorous Discovering the Symmetry and Interconnections Among the Platonic Solids
Coincidences are flashing lights alerting us to potential insights.

25. Question of the Day
What’s the most symmetric figure you can draw in the plane? What if you could only use straight lines?

26. What is a regular polygon?

27. The Five Platonic Solids
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron

28. Use your platonic solids to fill in the following chart:
Number of Vertices
Number of Edges
Number of Faces
Number of Faces at Each Vertex
Number of Sides of Each Face
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron

29. Observations
Look at the table. Do you notice any coincidences?

30. Duality
The faces of one correspond to the vertices of the other.

31. When you don’t know what to do, consider everything.
Section 4.6 The Shape of Reality How Straight Lines Can Bend in Non-Euclidean Geometries
When you don’t know what to do, consider everything.

32. Question of the Day
Is the shortest distance always a straight line?

33. How do we start to understand the geometry of something so large that it seems beyond our capacity to comprehend it? Start by looking at something similar.

34. Geometry of a Sphere

35. Geometry on a Saddle