Explore three-dimensional geometry, ZOME construction, and tetrahedron trigonometry in this 59-minute lecture. Delve into rational trigonometry concepts, including quadrance and spread, in both affine and projective settings. Learn about the ZOME system and its applications in geometric constructions. Discover new rational formulas for 3D trigonometry, covering side quadrances, face spreads, dihedral edge spreads, solid spreads, and quadrume. Examine specific examples like the right isosceles tetrahedron and regular tetrahedron, unveiling the metrical structure of these fundamental objects. Gain insights into the educational implications of rational trigonometry for mathematics teaching and explore challenges in ZOME constructions.
Overview
Syllabus
Introduction
The elusive tetrahedron
The basic quantities: quadrances and spreads
Quadrance between points
Spread between lines
Laws of affine rational trigonometry
Proof of main laws: Cross law
Proofs of main laws: spread law and quadrea
Proofs of main laws: Triple spread formula
Projective rational trigonometry
A projective triangle
Projective rational trigonometry
Projective quadrea
The ZOME construction system
ZOME quadrances
ZOME and spreads
Two coloured primitive ZOME triangles
Three coloured primitive coloured ZOME triangles
A ZOME tetrahedra
Tartaglia's volume formula
The Alternate spreads formula
The trigonometry of a tetrahedron
Dihedral spreads and quadrances
Solid spreads
Solid spread law
The right tetrahedron
Dihedral spreads relation
Dihedral spreads formula
Our example tetrahedron
Circumdiameter quadrance
Some ZOME challenges
Two key examples
The right isosceles tetrahedron
The regular tetrahedron
More info
Taught by
Insights into Mathematics
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