- More Introduction to course organization.
- Web Materials and Notes.
- What is Visual Mathematics?
- Mathematics that studies topics related to visual experience. [Geometry,
Topology, Motion]

- Visualization of mathematics that is not inherently visual. [Visualizing Counting]
- Example: Numbers...

- numeral: a symbol for representing a number
- Number: a form of universal language to describe anything/ physical things/ concepts related to measurement
- such as V, 5, five, cinq, chamesh, cinco
- Frege distinguished numerals from numbers in the late 19th century.
- We can compare numbers... for instance we say" 3 is less than 5"
- Is 3 smaller than 5?
- Numerals are symbols (visual or linguistic) that we use to represent numbers.
- We use numbers to measure (lengths) and put things in order (which was first).
- Another common visual representaion of numbers uses the number line.
- ___.___.___.___.___.___.___.______
- 1 2 3 4 5 6 7

Here the numerals are associated with points, so the points are considered to visualize the corresponding numbers.

[Use Wingeometry to illustrate numbers used for measurement.]

- We visualize equations that give relations between numbers with graphs in the coordinate plane.

3x + 2y = 6 is visualized by the graph of a line.

Use Wingeometry to visualize coordinates, points, and lines.

- Miscellaneous: Some topics we will study.
- The film lists as a guide to the course topics.
- Symmetry.(Chapter 5)
- Tiling. (Chapter 5)
- Similarity and Projection. (Chapter 4)

- The color problem. (Chapter 6)
- The motion problem.(Chapter 3)

- The Sphere and the Torus. Surfaces (Chapter 6)

Theme Question:Who first showed the earth
was a sphere?

- Discussion (brief) on the The Sphere
and the Torus.

- Coop work- 3-4 partners- record
responses of group. 5 minutes, followed by discussion.

Question 1. How can one ** distinguish
the sphere from a plane** (Flatland) based solely on experiences on the surface?

Question 2. How can one

- Measurement and the Pythagorean
Theorem (PT)

Show video on PT.

Outline:

Area of triangles = 1/2 bh

Area of parallelogram= bh

Scaling: a linear scale change of r gives area change of factor r^{2}.

3 questions: running, moat, wind power...

Proof of the PT: Similar right triangles:c=a^{2 }/c + b^{2 }/c.

applications and other proofs.

Prop. 47 of Euclid.

Dissection Proof.

Prop 31 Book VI Similar shapes.

Simple proof of PT using similar triangles of the triangle.

Use in 3 dimensional space