- More Introduction to course organization.
- Examples for Portfolios and Projects.

- Web Materials and Notes.
- Why focus on Visual Mathematics?
- Mathematics that studies topics related to visual
experience. [Geometry,
Topology, Motion]

- Visualization of mathematics that is not inherently visual. [Visualizing Counting]
- Classification: Triangles, circles: congruence and similarity. What common features are shared by circles and triangles?... They divide the plane into two sectors: "inside and outside". Not in the same class as the symbol. 8.

- Example of "visual Math" and counting:
- My Name: Martin Flashman
- How to start a letter to me: Hi ___________

- How many different openings are possible?

- We can visualize this problem with a "tree"
- This visualization allows us to count the possibilities
easily...
- There were several alternative countings suggested for this problem. This may be connected to some ambiguity in the statement of what would be an acceptable opening for the letter. So- if the assumption is made that some entry must be made then the choice of an omission for each selection would be eliminated from the tally, giving 25 possibilities. A distinction:
- "vague" meaning "Not clear in meaning or application." and
- "ambiguious" meaning "Open to more than one
interpretation.
*Ambiguous*indicates the presence of two or more possible meanings." - This is an example of a visualization used to understand and solve a problem that initially is not connected to anything visual .

Professor | Martin | Flashman |

Doctor | Marty | Flash |

Mister | Omit | Omit |

Omit |

seeing there are 9 possibilities for each of 4 title branches

so that the total is 9*4 = 36 possibilities.

- 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.

- 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.

- 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 on the The
Sphere and the Torus. [to be continued next class]

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.

To be continued next class!