Algebra and abstract structures · Maths EE idea · Solid

Roots of unity and regular polygons

A research question to start from

How do the roots of unity explain the products and sums of distances from one vertex of a regular polygon to the others?

A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.

Why it works as a maths EE

A neat geometric result proved by algebra, with room for your own generalisations.

Mathematics you would need

Complex numbers in polar form
Roots of unity
Factorising z^n − 1
Trigonometric identities

Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.

One possible line of attack

Prove that the product of the chord lengths from one vertex of a unit n-gon is n.
Investigate sums of squared chord lengths.
Generalise to points not on the polygon.

Scope and difficulty

Solid. Solid and self-contained.

Pitfalls

Proving only special cases.
Diagrams replacing algebra.

Where to start reading

Search a library catalogue or a university's open lecture notes for: product of chord lengths regular polygon roots of unity. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).

Make it your EE

Plan this EEOpens the free EE planner with this question as a starting point. Sharpen the questionSix quick checks for a focused research question. Self-check a draftCheck a section against each criterion, free.