3 years ago

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Geometry An Interactive Journey to Mastery
James S. Tanton, Ph.D.
Mathematician in Residence
Mathematical Association of America
https://www.thegreatcourses.com.
https://www.youtube.com/c/JamesTanton_SquineCosquineTanq/about.
https://en.wikipedia.org/wiki/James_Tanton.
The Elements is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria
https://en.wikipedia.org/wiki/Euclid's_Elements.
David Hilbert (1862 – 1943)
https://en.wikipedia.org/wiki/David_Hilbert.

ꕔThe second most-printed book ever.ꕔ

ꕔ Euclid’s ‘Elements’ was the first rigorous account of geometry (drawing on earlier, less rigorous works). Starting with a mere ten axioms (statements which cannot be reduced to simpler ones), Euclid defines points, lines, two- and three-dimensional spaces, and the mathematical objects which inhabit them. The lucidity of Euclid’s style and the tight logic of his proofs meant that ‘Elements’ remained one of the standard texts on geometry right up to the nineteenth century, until more rigorous definitions of geometry emerged, along with the discovery of non-Euclidean geometries. Over two thousand years after being written, such diverse people as Abraham Lincoln, Edna St. Vincent Millay, and Albert Einstein drew inspiration from Euclid’s mathematical masterpiece. ꕔ

ꕔ Euclid's Elements has been referred to as the most successful and influential textbook ever written.ꕔ

ꕔ Most of the theorems appearing in the Elements were not discovered by Euclid himself but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner. ꕔ

ꕔ The geometrical constructions employed in the Elements are restricted to those which can be achieved using a straight ruler/edge and a compass. Furthermore, empirical proofs by means of measurement are strictly forbidden: i.e., any comparison of two magnitudes is restricted to saying that the magnitudes are either equal, or that one is greater than the other. ꕔ

ꔷContent of the Elementsꔷ
1. Axioms of geometry, triangles, constructions
2. Computing with rectangles (products) and squares
3. Circles
4. Incircles and circumcircles of triangles, regular polygons
5. Proportions
6. Plane geometry, similarity
7. Number theory: divisibility, primes, gcd
8. Geometric sequences of integers
9. Infinitude of primes and perfect numbers
10. The irrationality of roots
11. 3D geometry
12. Cones, pyramids, cylinders, and volumes
13. Construction of the five Platonic solids

To be a Platonic solid, the tested shape must:
ꔷBe convex
ꔷBe three-dimensional (a polyhedron)
ꔷHave congruent faces
ꔷHave congruent corners (vertices).
Of the five Platonic polyhedra, the icosahedron most closely approximates a sphere.

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