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# DISSECTIONS OF AN OCTAGON

“Geometry” is a posh word for “shapes”. Here are two fun geometry puzzles for you, inspired by a question form the UK Mathematics Trust Challenge. Anyone can play, even non-mathematicians:

1. The first octagon has been cut into four congruent (same size, same shape) quadrilaterals (four sides). Can you figure a way to dissect the octagon into four congruent pentagons (five sides)? How about into four congruent hexagons (six sides) – or even heptagons and beyond?
2. What fraction of the red-and-white octagon has been shaded red? (and can you prove it?)

# Or: a beginner’s guide and glossary for the 4th spatial dimension.

INTRODUCTION: imagine a tightrope walker: she can only change her position in one direction: forwards & backwards, so we need just one number – how far along the rope she is – to specify her position. That’s one dimension!

Now picture an ant crawling on a tabletop: the ant can crawl forwards & backwards or left & right, and we now need two numbers (e.g. the ant’s horizontal and vertical distance from a particular corner of the table) to pinpoint exactly where the ant is. That’s an extra degree of freedom: two dimensions!

And finally, picture yourself wearing a jetpack: you can now move freely in all three of our spatial dimensions: forwards & back, left & right, AND up & down. Three numbers are now required to describe your position at a given time e.g. latitude, longitude and altitude. That’s three dimensions!

If you think that sounds fun, imagine how cool it would be if there was a fourth, new direction in which you could also travel! What would 4-dimensional space look like, and what sort of shapes would inhabit it?

# Or “how maths can increase the dramatic tension of Reality TV competitions”.

SUGGESTED LEVEL: UPPER SECONDARY

Below is a hierarchy of increasingly dramatic (and mathematically complicated) systems for a reality TV competition such as “Britain’s Got Talent” or “Strictly Come Dancing” to announce their competition winner. Just one rule is needed for creating each system of announcements from the previous one: in mathematics this concept is referred to as RECURSION.

#### RECURSIVE STEP: to create the next level, we replace each single announcement of the previous level with a set of announcements of all the other contestants!

Confused? Let’s see how this would work in practice:

# WHY DO HEXAGONS OCCUR NATURALLY IN NATURE?

This is Giants Causeway on the North Coast of Northern Ireland, famed around the world for its awe-inspiring hexagonal stone plinths. Incredibly, the stones are built not by a genius mathematician or engineer but by mother nature. In this article I’ll explain how it is that nature can afford us such a beautiful display.

The plinths here are in fact igneous basalt columns, created when molten lava comes up from inside the earth and cools. As the rock cools it contracts, and this changing shape means that as it solidifies the rock must crack to release the pressure (similar to the way that an ice cube warms, contracts and cracks when you put it into your drink).

# House of Maths School Workshops Primary & Secondary in Dorset & South - WHAT’S SPECIAL ABOUT THE NUMBER TWO?

## WHAT’S INTERESTING ABOUT THE NUMBER $2$?

### TWO is the only even prime number, and the number of Ronnies in the famous duo. It’s the sides on a digon (two-sided shape such as the panels of a juggling ball), and it’s what it takes to tango. It’s a dual, a duel, a duo and a twin. Two has two homophones: ‘to’ and another one ‘too’.

Here are a couple of pairs of deuces:

CONTINUED FRACTIONS: consider this fraction:

# $\frac{2}{2+\frac{2}{2+\frac{2}{2+…}}}$

If we continue the process indefinitely, what answer do we get? The answer is $-1+\sqrt 3$ (which is irrational i.e. cannot be written as a integer fraction !). To see this, call the continued fraction $x$ and observe that $\frac{2}{2+x}$ gives the same continued fraction as the original $x$. Now rearrange $\frac{2}{2+x}=x$ and solve the resulting quadratic (degree TWO) equation.
What number would this continued fraction give? $\frac{3}{3+\frac{3}{3+…}}$

TWO is the wheels on a bicycle, the ends of a diameter or a diagonal, and two is a duplicitous timer. We need two co-ordinates to find our place on a map in flatland; even at sea on a sphere just two numbers – latitude and longitude – tell us where we are. Not only the Earth’s surface but any plane in fact has two ‘dimensions (a curve has one and a point zero dimensions).

The number two is in a sense indifferent to ADDITION, MULTIPLICATION and EXPONENTIATION, since 2 plus 2 = 2 times 2 = 2^2; swap three for two and it’s no longer true.
$2+2=2\times 2=2^2$

BINARY CODES: these are codes made of two ingredients, such as Braille (raised and flush), Morse (dots and dashes), Bar Codes (black and white stripes), QR codes (black and white squares) and of course Base two (0’s and 1’s).

HERE’S A DILEMMA: there are of course TWO answers to this “missing word” puzzle: can you find them both?

Number two: we salute you!

# House of Maths School Workshops Primary & Secondary in Dorset & South - WHAT’S SPECIAL ABOUT THE NUMBER ONE?

### WHAT’S INTERESTING ABOUT THE NUMBER ONE?

ONE is the only number that’s the same in binary, base 10 and Roman Numerals.

It’s the wheels on a unicycle, the rails on a monorail and the players when you go solo.

One is the first odd number, the first triangular, square, pentagonal and hexagonal number, and the first tetrahedralcube and Fibonacci number.

It’s the only whole number whose letters O-N-E are in reverse alphabetical order (but which is the only whole number whose letters are in forward alphabetical order?).

One is the only positive integer (whole number) which is neither prime (exactly two factors: one and itself) nor composite (more than two factors).

NORMAL ARITHMETIC:
The number one is used to make up all the other integers:
1=1
2=1+1
3=1+1+1 etc

AXIOMATIC SET THEORY (von Neumann ordinals) :
The number zero is used to make up all the other integers:
0=∅={} (the empty set)
1= {Ø}={0}
2={Ø, {Ø}}={0,{0}}
3={Ø,{Ø},{Ø, {Ø}}}={0,{0},{0,{0}}} etc

Axiomatic set theory started in the early 1900’s as an attempt to place mathematics on a firm logical footing free of paradoxes. For further reading, search Wikipedia for Russell’s Paradox, Cantor’s theorem, cardinal numbers, ordinal numbers, put aside about a month of reading time and get ready for an incredible ride!
Which definition of the integers do you prefer: using ones or zeroes?

One is gender-neutral: one is so pleased you are reading one’s post!

So. number one, we salute you: you are singularly unique and one of a kind. A big hand for NUMBER ONE!

# House of Maths School Workshops Primary & Secondary in Dorset & South - DOGARITHMS

### WHAT’S THE POINT OF LOGARITHMS?

On meeting this adorable litter of ten Dalmation Puppies the other day, I quickly spotted (ahem) the need for Dogarithms – the canine equivalent of logarithms. The conversation with dog breeder Maxine went something like this:

# House of Maths School Workshops Primary & Secondary in Dorset & South - HOW MOST PEOPLE CAN BE “BETTER THAN AVERAGE”!

### JANE: “I did better than average, but then so did most of the class”

Jane’s statement seem ridiculous – how can most people be better than average?  Using some mathematical trickery, here’s how she could be logically correct.