Introduction to Geometry
In this chapter we review some well-known propositions of elementary geometry, stressing the role of symmetry, referring to Euclid’s propositions by his own numbers — used worldwide for more than two thousand years. Since F. Commandino (1509) translated Archimedes, Apollonius, and Pappus, many theorems in the same spirit have been discovered and studied in detail; as the present tendency is to favor other branches of mathematics, we mention only a few that seem particularly interesting.
Core structure
- Triangles
- Euclid
- Primitive Concepts and Axioms
- Proposition 1.25
- Proposition 1.26
- Pons Asinorum
Main ideas
- The review stresses symmetry while surveying elementary geometric propositions.
- Euclid’s work “will live long after all the text-books of the present day are superseded and forgotten.”
- The text invokes the problem of definitions — “When I use a word, it means just what I choose it to mean,” Humpty Dumpty (Lewis Carroll).
- It restates the parallel postulate: two lines making interior angles summing to less than two right angles will, if extended, meet on that side.
- A worked theorem: given triangles with corresponding equal sides and an equal extension, the corresponding cevians are equal (
AD = A'D' ).
Why it matters
The book supplies the classical-geometry foundation — symmetry, Euclidean axioms, and theorem-proving in the traditional spirit — against which Fuller’s synergetic and geodesic geometry can be read and compared.
See Also
- Geodesic Math and How to Use It — applied geometry for geodesic and tensegrity structures
- Synergetics — Fuller’s own whole-system geometry
- Dome Cookbook of Geodesic Geometry — applies this classical geometry to the practical derivation of geodesic domes
- George W. Hart — sculptor-geometer continuing the polyhedral tradition this text grounds
- Euclid — the classical geometer whose Elements this modern survey carries forward
- Pythagoras — origin of the number-and-form tradition surveyed here
- Archimedes — the semiregular solids treated in this survey
Sources
- introduction_to_geometry/ — book project directory (repo-local source tree)
- introduction_to_geometry/index.md — synthesis index for the source project