Mathforge – Counting On My Fingers

loopspace

2018-10-22

The Counting On My Fingers section of my website is my writings on Mathematics. I frequently find myself exploring different parts of Mathematics and finding out stuff that I didn't know before. Rather than try to keep it all in my head, or to just allow it to fade, I like to write down what I learn. This also helps me to make sure that what I think I think is what I actually think – without the step of writing something down, I can fool myself into thinking that I know what's going on without actually doing the hard work. Once it's written down, why not make it public?

## 1 Recreational Mathematics

The posts in this section are, in the main, inspired by playing with mathematics.

1. Euler's equation consistently tops the list of "most beautiful equation". I think that this is ill-deserved. This is my explanation as to why.

2. A little playing with Euler's identity reveals a property of the Golden Ratio:

 $2\mathrm{cos}\left(i\mathrm{log}\left(1+\varphi \right)\right)=3$

Playing with this idea leads to some fun with Diophantine equations and recursive sequences of similar identities.

3. My excursions into mathematically-inspired art couldn't go far without encountering my calligraphic interests.

4. This is my contribution over the semi-annual debate as to which angle deserves a symbol of its own. There can only be One True Right Angle.

5. Whilst in Norway, I came across an interesting article shedding new light on the Vikings' approach to mathematics1.

1Just in case it's not completely obvious, look at the copyright notice at the foot of the page of this article.

7. This is my explanation of the mathematics behind the game Dobble. Other explanations exist on the web …

8. A weekend spent struggling with practical geometry led to this investigation into the field of Inverterbar Komplekse Entydig Analytisk Matematikk, or IKEA Matematikk for short.

9. I recently came across a fact about the Fibonacci sequence that I'd not encountered before. I'm not the biggest fan of the Fibonacci sequence so my immediate reaction to this was to look for a more general picture to fit this into. This is that general picture.

10. I recently came across Triskele Globes. After making one, I noticed that the fold didn't look as though it was in quite the right place. There seemed to be a couple of points of deformation, particularly at the corners. So I thought I'd have a look at figuring out what the curve ought to be.

11. The author of the Math with Bad Drawings posted an interesting article about a combinatorial interpretation of Fermat's Last Theorem. It got me thinking. This is the result of those thoughts.

12. A puzzle posted on twitter by https://twitter.com/Cshearer41 caught my eye as it seemed to be about braids. Turned out it was also about bells and taxicabs.

13. Some thoughts on learning that people have used Voronoi Diagrams to study the growth of forest canopies.

## 2 Similar Triangles

It's my contention that all of school geometry can be traced back to properties of similar triangles.

1. I recently came across a method of solving quadratic equations using a ruler and compass. The construction is so simple that I couldn't take it at face value and had to figure out for myself why it worked.

2. There are many, many proofs of Pythagoras' Theorem and amongst them I've found one that I particularly like. It has ingredients that closely relate to what Pythagoras' Theorem is really about (how lengths behave in Euclidean Space), and extends to a proof of the cosine rule.

3. Having come across an elegant proof of the double angle formula for sine which revolves around area, I wanted to see if there was an extension to the compound angle formula.

4. I read a post about finding Pythagorean triples. A few things about it intrigued me, and on digging through the details I found that yet again, similar triangles had a role to play.

5. I've become a little obsessed with similar triangles and the latest symptom of it is with finding diagrams for any given trigonometric identity (rather than proving them by a series of algebraic manipulations).

6. Yet more on similar triangles, this time looking at a characterisation of a circle that crops up when studying loci with complex numbers – though no knowledge of complex numbers is needed to understand it.

## 3 Beyond the Syllabus

These posts are directly inspired by teaching topics (other than similar triangles).

1. Here's an alternative take on the geometry of completing the square, devised as a way of finding the width and height of a rectangle from its perimeter and area. As such, the method is suitable for students before they've formally met quadratics.

2. This is an end of term or end of topic activity using quadratics to draw out a familiar shape.

3. I originally wrote this way back in 2011 for a course where students were required to prove results rigorously. I used it as the basis for this answer on Mathematics.SE, and so am reposting it here.

## 4 Problems and Puzzles

Every now and again I'll attempt a mathematical problem. If I learn something, there's a chance I'll write it up.

1. There's something about mathematical puzzles that I find not quite to my liking. Two puzzles that recently came my way gave me the opportunity to try to figure out why that is.

2. Having decided why I don't like mathematical puzzles as puzzles, I decided to have a go at one thinking as a mathematician. This is my – quite lengthy! – walk through of that thinking.

3. By taking a systematic approach to the Four Fours puzzle, it's possible to find ways to vary the puzzle when setting it to add a little variation.