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 Identity.

   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. Golden Numbers.

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

   $$2\cos (i\log (1+\varphi ))=3$$

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

3. Mathematical Calligraphy.

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

4. The Right Angle.

   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. Viking Mathematics.

   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.

6. Pieces of Math.

   I made a poster to show that mathematics is made up of many areas. You can download it from here.

7. Fields and Games

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

8. IKEA Matematikk

   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. Adding up Fibonacci

   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. Curving a Triskele Globe

    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. Counting Fermat

    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. Bells, Braids, and Taxicabs (Chalkdust Article)

    A puzzle posted on twitter by Catriona Shearer caught my eye as it seemed to be about braids. Turned out it was also about bells and taxicabs.

13. Voronoi Trees

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

14. Intermediate Axes

    This is a follow-up to a claim I made in a comment on The Aperiodical that any reasonable shape in the plane has two orthogonal diameters that are equal.

15. Absolute Batman

    This follows on from the post Colouring with Quadratics in which I describe how to use quadratics to draw a Batman logo. In this post, on an idle whimsy, I show how to use the absolute value function to turn a shape defined by multiple paths into a single parametrised curve.

16. Squircle Clock

    I got an idea for turning a time into a shape using Fourier series. It's a bit silly, but it's fun.

17. Colouring Origami

    This post combines two of my hobbies: Mathematics and Origami. It is inspired by trying to make a modular origami polyhedron with not enough pieces to do it with a single pattern of paper. Given the number of patterns available, and how many I could use of each one, I wanted to distribute the choices with some degree of symmetry about the polyhedron. This leads to a few interesting colouring problems, as outlined in this post.

18. Colouring More Origami

    Some origami modules have a curious property that the colour of the module is not shown on the edge to which it corresponds. In this post, I took a look at these to see how to make them still appear to have a coherent colouring. This led to a nice occurrence of ribbon graphs, something that I encountered many years ago when researching in algebraic topology.

19. Chords and Tangents

    In a post on BlueSky, Dave Jones asks:

    Hi #Alevelmaths, tangents are currently being discussed with first years. I have a question: for which functions are the two statements logically equivalent, "$f(x)$ has a tangent that crosses at more than one point" & "the second derivative takes +ve and -ve values"? Thanks in advance.

    I had a go at answering, but short message systems, like BlueSky, are not a great medium for developing a mathematical argument and checking that it holds tight. So I figured that as I've done the thinking, I may as well get it written down somewhere.

2 Similar Triangles

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

1. Solving Quadratics with a Ruler and Compass.

   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. My Favourite Proof of Pythagoras' Theorem.

   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. An Area Proof of the Compound Angle Formula.

   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. Finding Pythagorean Triples with Similar Triangles.

   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. Trigonometric Identities

   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. That Which We Call a Circle

   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.

7. Similarity and Area

   This is a summary of my thinking about Pythagoras' Theorem and how it relates (or not) to the concept of area. It also details my attempts to find an area-based proof of the Intersecting Chords Theorem.

8. Dissecting an Annulus

   Some more thinking about Pythagoras and, in this case, its interaction with the classic area of an annulus problem. (For more on that problem, see also The Area of an Annulus).

9. My Second Favourite Proof ...

   … of Pythagoras' theorem. A simple proof using similar triangles and with an extension to the cosine rule.

10. The Proof that goes on Proving

    I had occasion to revisit My Favourite Proof of Pythagoras' Theorem and discovered that it had more to it than I'd originally realised.

3 Beyond the Syllabus

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

1. Quadratics and Rectangles.

   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. Colouring with Quadratics.

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

3. On the Nature of Proof (Original Version).

   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. Cubic Footsteps.

   I recently figured out for myself the crucial step in solving a cubic equation. Perhaps not all that exciting, but it was momentous enough for me to feel it worth writing down before I forget it again.

5. Reflecting on Rotations.

   I saw a discussion on whether the operation $x\mapsto -x$ should be viewed as a rotation or a reflection. When one considers that $x$ might be a function, the story gets quite interesting.

6. Chain Rules Supreme.

   The chain rule is the most important of the three differentiation rules that are taught at A level. But the way I know of deducing the product rule from the chain rule uses partial derivatives so isn't very suitable for using in a regular A level lesson. So I was pleased to figure out a way to deduce it that doesn't require anything not in the A level. It just involves a sneaky relationship between squaring and multiplying.

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. A Tale of Two Puzzles.

   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. Find the Angle.

   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. The Four Fours Puzzle.

   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.

4. A Digit Sum Puzzle

   Sometimes a puzzle deserves a second look.

5 Scaffolding

A Twitter conversation with Catriona Shearer led to the idea of exploring not just how one might solve a puzzle but also how the setter came up with the original puzzle.

1. Congruent Rectangles.

   In this post, Catriona and I explore how she came up with the Congruent Rectangles puzzle and then how I devised a picture proof of it.

2. The Area of an Annulus.

   This post looks at a puzzle of Catriona's based on the classic derivation of the area of an annulus. We look at how Catriona adapted it to create the puzzle and how I found a generalisation of it.

3. Squares in a Circle.

   This was a curious puzzle which sparked quite a considerable discussion. It started with two squares sitting inside a circle and led through various permutations by a variety of people to quite a different place. Catriona also found several related puzzles from that same starting point.

4. The Birthday Parallelogram.

   This is the story of the Birthday Parallelogram, a puzzle in which the answer came first, and the puzzle afterwards.

5. The Mystery Puzzle

   Catriona found a puzzle in her notebook that she hadn't posted, however it had been sufficiently long ago that she'd created it that she couldn't remember her thinking at the time. So rather than exploring how she came up with it, we thought it would be interesting to find out what it was like for her to do one of her own puzzles!

   It's also a fabulous example of the Agg Invariance Principle.