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Three dimensional geometry, ZOME, and the elusive tetrahedron

Lectures notes at geometry of three dimensional space, despite its obvious importance, is a sadly neglected topic in modern mathematics. One of the reasons is that the topic is rather awkward and difficult when approached with the standard tools of metrical affine and spherical geometry. In this talk we show that ideas of rational trigonometry, both in the planar and spherical (elliptic) setting, open new doors to our understanding, supported by fascinating algebraic relations. We will start with a quick linear algebra presentation of rational trigonometry, introduce the beautiful and remarkable ZOME construction system, and then tackle the trigonometry of a tetrahedron---the fundamental object in three dimensional geometry. We introduce the basics of rational trigonometry, first in the affine and then in the projective setting, using the notions of quadrance and spread instead of distance and angle. (Quadrance can be thought of as the square of the distance, and spread the square of the sin of an angle; both notions come in an affine and a projective flavour.) The main laws of affine rational trigonometry, namely the Cross law, Spread law and Triple spread formula are proved using only a bit of first year linear algebra and familiarity with the dot product. All maths teachers/lecturers who currently teach trigonometry might like to have a think about the educational implications of this first fifteen minutes.The projective formulas will be familiar to those following the UnivHypGeom series. The main object is a projective triangle, or tripod, consisting of three concurrent lines in 3D space, and the corresponding three planes formed by pairs of those lines. The spreads between the lines are called the projective quadrances of the tripod, while the spreads between the normals to the planes are the projective spreads of the tripod. The projective relations between the three projective quadrances and three projective spreads are deformations of the planar laws.The Zome construction system is introduced and we exhibit the primitive Zome triangles, and a particular tetrahedron derived from a dodecahedron and one of its faces.The heart of the talk is the derivation of new rational formulas for three dimensional trigonometry, involving side quadrances, face spreads, dihedral edge spreads, solid spreads at the vertices, and 144 times the square of the volume, which we call the quadrume. These laws are not at all complete, in the sense that there are other relations and further secondary quantities too which we do not discuss.We illustrate the main laws with explicit concrete formulas for not only our example tetrahedron, but also the right isosceles tetrahedron (analog of the 90/45/45 triangle), and then the regular tetrahedron. It seems remarkable that the metrical structure of this fundamental object is laid out for the first time here, on YouTube, in 2012!This talk was given to the Pure Mathematics Seminar of the School of Mathematics and Statistics at UNSW on July 31, 2012. Thanks to Michael Cowling, David Hunt, Michael Reynolds and Thomas Britz for the interesting questions, and thanks to Nguyen Le for help with the filming.
Length: 58:39


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