# ET7246 - Geometry and Trigonometry Dr. David R. Wilkins

## Material for Hilary Term 2021

Resources for the study of Euclid's Elements of Geometry
An account of an investigation of the Regular Pentagon using Complex Numbers
This account begins with a description of Sir William Rowan Hamilton's formulation of the foundation of the complex number system in which he considers a complex number to be an algebraic couple consisting of an ordered pair of real numbers, with operations of addition, multiplication, subtraction and division of such algebraic couples defined in a suitable fashion. The account continues with an investigation of those complex numbers z that satisfy the equation z5 = 1. The five roots of this equation are 1, ω, ω2, ω3 and ω4, where the real and imaginary parts of the complex number ω have values cos(2π/5) and sin(2π/5) respectively. The investigation that follows determines the values of the Cartesian coordinates of the vertices of a regular pentagon inscribed in the unit circle centred on the origin on the plane, where one of the vertices of that pentagon is located at the point with Cartesian coordinates (1,0). The formulae thus obtained are expressed in terms of the value of the Golden Section. A result is then deduced which ensures that a right-angled triangle can be formed whose sides are respectively equal to the sides of a regular decagon, a regular hexagon and a regular pentagon all inscribed in the same circle. This result in in fact Proposition 10 in Book XIII of Euclid's Elements of Geometry. This account concludes with some historical remarks concerning ancient Greek investigations concerning the five Platonic solids.
Further Miscellaneous Resources for the study of Geometry and Trigonometry

## Material for Hilary Term 2018

Leaving Certificate Mathematics Syllabus
The syllabus for Foundation, Ordinary and Higher Level, for examination from 2015, and incorporating the document Geometry for Post-primary School Mathematics, the preparation of which was undertaken principally by Anthony O'Farrell, with assistance from Ian Short.
An Overview of Geometry for Post-primary School Mathematics in Ireland
This document is an evolving document, representing work in progress, and is likely to be subject to constant revision whilst teaching of Module ET7246 is in progress.
Axiomatic Foundations of Planar Geometry
This document is an evolving document, representing work in progress, and may be revised whilst teaching of Module ET7246 is in progress.
Selected Circle Theorems
A sequence of presentation slides incorporating circle theorems, and concluding with a discussion of “golden sections” and “golden triangles”
, establishing the theoretical background underlying the construction of the regular pentagon using straightedge and compass. (This series of slides stops short of explaining and justifying the construction of the regular pentagon itself.
The Pentagram
A sequence of presentation slides concerning properties of the pentagram formed within a regular pentagon, focussing in particular on the equality of all the small angles in the figure, and the properties of the various “golden triangles” that emerge from the construction.
Trigonometry
A sequence of presentation slides concerning trigonometry
A GeoGebra construction of a regular pentagon using straightedge and compass

## Resources for the Study of Geometry and Trigonometry

Resources for the study of Euclid's Elements of Geometry
Resources for the study of the history of geometry and trigonometry
Resources concerning the teaching of geometry and trigonometry
Miscellaneous resources for the study of geometry and trigonometry
Axiom Systems for Synthetic Geometry