MAS31004 INDIVIDUAL PROJECT (PURE & HISTORICAL)
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MAS31004 INDIVIDUAL PROJECT (PURE & HISTORICAL)
Overview. You will carry out independent research on Euclid’s Elements, and apply your mathematical problem solving ability and understanding of the ax- iomatic method to unfamiliar settings. You will first work through and interpret unseen construction problems in Questions 1 and 2. This is followed by a more modern approach to axiomatic geometry in Question 3, where you will be re- quired to create and explore models of new geometries and consider questions about (in)equivalences of geometric axioms. The project culminates with Ques- tion 4 which will require you to interpret, understand and explain the significance of Elements and its influence on your degree programme.
To do well. Your answers will need to clearly explain and communicate your understanding and insight to the relevant audience (see below).
Submission. You will submit your work before 16:00 on Wednesday the 13th of December via the link on Blackboard.
Question weighting. Please note that the essay question (Question 4) is consid- ered to be the most important question on this project. It will carry the majority of marks. So the majority of your effort should be spent on this question.
Audience and format. Your answers should be understandable to any third year student on your degree programme. Your submission should be written up in LaTeX, Microsoft Word or an equivalent word processing programme. You should use a consistent referencing style, and follow the university guidance on plagiarism. You may use any format for diagrams (including hand-drawn) and correctly attribute the source for any diagrams which you do not make.
Question 1. Let ∠AOB be a given angle of size π/7.
(a) Construct a point X so that OX trisects ∠AOB . Your answer should clearly lay out the steps of the construction and contain a rigorous proof that your construction is correct.
(b) Give a very brief explanation of why your answer to Part (a) does not contradict Pierre Wantzel’s 1837 resolution of the trisection problem.
Question 2. The argument below shows a construction in the spirit of Euclid.
To construct a perpendicular to a line AB through a point P on AB.
Choose C not on AB.
Draw a circle Γ centred at C of radius |CP| .
Extend AB if necessary to intersect Γ at a point Q.
Draw QC.
Extend QC until it intersects Γ at D.
Draw DP.
The line DP is perpendicular to AB.
(a) Clearly justify each step in the above argument by making reference to the axioms and propositions used from Euclid’s Elements, and highlight steps that Euclid might have been unhappy about.
(b) Euclid did not included this construction in Elements. Offer a brief expla-nation for why you think this construction was omitted.
Question 3. We will define a “MAS31004 geometry” to be a geometry which satisfies Hilbert’s incidence axioms and Elements Book I Proposition 31. That is,
a “MAS31004 geometry” is a geometry that satisfies the following four axioms.
I1: There exists a unique line passing through any two pairs of points.
I2: There exists three non-colinear points.
I3: Every line has at least two points.
P: If two distinct lines L and M intersect then there does not exist a line which is parallel to both L and M.
Answer the following questions about MAS31004 geometry.
(a) Give an explicit example of a MAS31004 geometry with infinitely many points and give an example of a MAS31004 geometry with finitely many
points. Your answer should fully justify why the four axioms are satisfied.
(b) Show that Axiom P is independent of Axioms I1, I2 and I3.
(c) In a MAS31004 geometry, is Axiom P stricly weaker, strictly stronger or equivalent to the following statement:
“If L is a line and p is a point not on L then there exists a unique line passing through p which is parallel to L.”
You should fully justify your answer.
(d) Is the statement from Part (c) equivalent to P in a geometry which satisfies
Euclid’s first four postulates? You should fully justify your answer.
4. Write a short account of at least one of the ways that Elements
Book I has influenced your undergraduate degree programme.
Your answer should:
. Be no longer than 1200 words (captions and displayed mathematics will not count towards the word count).
. Use explicit examples and references, and use a consistent referencing style.
. Include an appendix with a completelist of modules (names and code) that you have taken at Sheffield.
2023-11-29