Notes on Euclid Elements

Euclid pdf in Greek and English

David Joyce, Clark University: Eudlid html with Commentary

geogebra.org: Drawing Tool simulating compass and straight-edge

pokerface scripts

• i2d.py
• ispiral.py
• liner.html

curriculum python scripts

• example_sinewave_interactive.py
• helix3.py - renamed
• ispiral.py
• test3ddraw.py

khan academy - vector dot product and vector length
khan academy - matric vector products
khan academhy - vectors and spaces - defining the angle between vectors
ikhan academy - electric motors - the dot product

Gil Strang lecture 1: The column space of A contains all vectors Ax

tibetan buddhist chant
https://www.youtube.com/watch?v=iei5QA_aPp8

Projects

• Beauty patterns
• Model sk8 actuators
• Linear arithmetic algebraic geometric exponential logarithmic
• Patterns earthquakes
• String ball not tangle tangled knot skein
• Normal Pareto

Mauy in Phuket https://www.thephuketnews.com/mauy-the-graffiti-artist-spraying-a-wall-near-you-73183.php

Street Art in Chiang Mai http://www.hipthailand.net/variety/art-design/624

Chomsky
The capacity for language and for mathematics did not come about through natural selection. https://youtu.be/1X-AkJZUIiE

Explanation of complex numbers https://youtu.be/ALc8CBYOfkw

Tesla battery plant in India https://www.scmp.com/news/asia/south-asia/article/3123802/india-promises-tesla-lowest-production-costs-will-persuade

Sapolsky: three brains: reptilian, limbic, neo cortex https://youtu.be/hg6XUYWj-pk

George Carlin: sociology: individuals versus groups https://youtu.be/Y2NjUKOw1Qg

Chomsky on merge function

• Combine two objects into bigger one
• Example of language used as thought

https://youtu.be/9JScy7ulDpE

Looking for a variation of multi-armed archemides spiral https://math.stackexchange.com/questions/3718071/looking-for-a-variation-of-multi-armed-archemides-spiral

Who benefits from these beliefs?

If language has not evolved, per Chomsky, can the same be said for all aspects of human behavior?

Each proposition describes how to draw a geometric shape using only a straight-edge and a compass,

proving equality and inequality of lines and angles, without explicit measurement of either.

Each proposition builds on the ones before.

And therefore, each proposition is an axiom.

2D shapes that lie in a single plane. Polygons.
point, line, plane, surface
angle: acute, obtuse, perpendicular
boundary, figure: circle, rectilinear
circle, center, semicircle
rectilinear figures: trilateral, quadrilateral, multilateral
trilateral: equilateral, isosceles, scalene, right-angled, obtuse-angled, acute-angled
quadrilateral: square, oblong, rhombus, rhomboid, trapezia
parallel lines (infinite)

Proposition 1.1. How to construct an equilateral triangle.

• Draw the base of the triangle as a line segment $\overline{AB}$.
• Use a compass to draw a circle with center $A$ and radius $B-A$.
• Draw a second such circle with center $B$.
• Take the point $C$ where the two circles intersect.
• The triangle $ABC$ is equilateral.

Proposition 1.2. How to draw two line segments of equal length.

Proposition 1.3. Same as 1.2, alternate method.

Proposition 1.4 A proof that two triangles with two sides and the intervening angle equal, are equal triangles.

Proposition 1.5 A proof that in an isosceles triangle, the two angles at the base are equal.

Proposition 1.6 A proof that in a triangle having two angles equal, the opposite sides will be equal also.

Proposition 1.7 A proof that for any three points, there is only one triangle.

Proposition 1.8

Proposition 47. A proof of the Pythagorean theorem.

10 of the 14 propositions can be restated algebraically. quadratic equations.

Proposition 1. The distributive property of multiplication over addition.

\begin{align} x(y_1+y_2+\cdots+y_n) = xy_1+xy_2+\cdots+x y_n\\ \end{align}

Euclid did not use the word “distributive”, nor did he use this equation.
He told the story in terms of lines and rectangles.
x is the length of a line. xy is the area of a rectangle.

Proposition 2. Same as prop 1, but only using 1 cut, two rectangles.

\begin{align} x(y_1+y_2) = xy_1+xy_2\\ \end{align}

Proposition 3.

\begin{align} x = y + z \iff xy = y_2 + yz\\ \end{align}