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ET7246 - Geometry and Trigonometry
Dr. David R. Wilkins
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Material for Hilary Term 2021
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- 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
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.