Thursday, January 20 (ppt)
 More Introduction to course organization.

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 there numerals are connected to points, so the points are
considered
to visualize the corresponding numbers.
Use Wingeometry?
 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? or Winplot?
 Another example of "visual Math":

How to start a letter to me: Hello ___________
Professor  Martin 

Doctor  Marty  Flashman 
Mister  Flash  omit 
Omit  Omit 


How many different openings are possible?
 We can visualize this problem
with a "tree"

This visualization allows us to count the possibilities easily...
seeing there are 8 possibilities for each of 4 title branches
so that the total is 8*4 = 32 possibilities.
 This is an example of a visualization used to understand and
solve a problem
that initially is not connected to anything visual .

Miscellaneous: Some topics we will study.

The film lists as a guide to the course topics.

The color problem.
 The motion problem.

The Sphere and the Torus.
Who first showed the earth was a sphere?

Measurement and the Pythagorean Theorem (PT)
a^{2 }+ b^{2 }= c^{2}

Measuring angles, lengths and areas.

Squares, rectangles, parallelograms and triangles.

Dissections, cut and paste methods of measurement.

Cutting and reassembling polygons.

The Square Me Puzzle.

Do Pythagorean
Activity Sheet

Show video on PT

Puzzles and Polygons

Flatland and the plane

The triangle, quadrilateral, pentagon, and hexagon.

More on measurements of angles and areas of polyons.

Activity: Measuring angles in regular polygons.