======Notes on Euclid Elements====== [[http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf|Euclid pdf]] in Greek and English David Joyce, Clark University: [[https://mathcs.clarku.edu/~djoyce/elements/elements.html|Eudlid html]] with Commentary geogebra.org: [[https://www.geogebra.org/calculator| Drawing Tool]] simulating compass and straight-edge\\ ====Notes==== 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? ====Contents==== ^ Book ^ Title ^ Definitions^ Propositions^ | 1 | Fundamentals of Plane Geometry Involving Straight-Lines | 23| 48| | 2 | Fundamentals of Geometric Algebra | 2| 14| | 3 | Fundamentals of Plane Geometry Involving Circles | 11| 37| | 4 | Construction of Rectilinear Figures In and Around Circles | 7| 16| | 5 | Proportion | 18| 25| | 6 | Similar Figures | 4| 33| | 7 | Elementary Number Theory | 22| 39| | 8 | Continued Proportion | 0| 27| | 9 | Applications of Number Theory | 0| 36| | 10 | Incommensurable Magnitudes | 16| 115| | 11 | Elementary Stereometry (Solid Geometry) | 28| 39| | 12 | Proportional Stereometry (Measurement) | 0| 18| | 13 | The Platonic Solids (Regular Solids) | 0| 18| | | | 131| 465| 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. ====Book 1. Plane geometry.==== 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)\\ {{ :euclid_propi1.gif?300|}} 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. ====Book 2. Geometric algebra.==== 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}