AQUILA Magic Maths

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MARCH 2020

B i g i d ea s for i n q u i s i t ive k i d s

Magic Maths

e to our m o lc e w d n a Hello

e u s s i s h t a M c i Mag , of AQUILA e issu

Numbers in nature Numbers don’t just exist for humans, you know. Bear witness to some particularly savvy cephalopods, here.

iraculous and what a m it is too.

Shaken not stirred: Aquila peers into the past at the life and times of the original 007, John Dee, wizard of Tudor England.

Numbers are powerful.

Humans have been doing maths for thousands of years but, if I am completely honest, numbers make me anxious. They always have. I consider myself a Word Person. Wherever I see lots of numbers on a page together I feel the walls begin to close in on me ever so slightly – especially if I am expected to use those numbers to actually do something.

For a long time I used my anxiety about numbers to hide from them whenever possible. I made the excuse that ‘numbers really aren’t my thing’ and asked other people to handle them for me, believing that to leave me in charge of anything as important and challenging as maths, would be irresponsible. Possibly even DANGEROUS.

Chinese Numerology ‘Whooooah there’, I hear you say, ‘AQUILA – my favourite magazine in the universe – has abandoned the teachings of science in favour of weirdy beardy mystical hoohah.’ Well, no.

Not quite. Numerology might not be supported by science, but it HAS shaped the world we live in. Let’s find out how.

And do you know who changed my mind about all that? You lot did, of course! Every day we receive lots of lovely letters from AQUILAnauts and it turns out a lot of you love to do maths.

Your positivity and confidence is infectious. It has shown me that maths isn’t just about scary numbers. It’s actually about finding patterns and connections, representing those connections, and using those representations to do fun new stuff. I mean… isn’t that also what writers do with words, what painters do with paint and what sculptors do with stone? I don’t know about you, but I think that is pretty magical! Thank you for showing me that maths can be for everyone!

Freya Hardy, ed We hope you have a momentous March. Don’t forget to join us back here in April for our excellent Earth issue. It’s going to be out of this world.

Houdini think you are? Polly makes a great escape – vaudeville style.

Rock around the clock (it ’s more like a calendar than a clock. Just saying, ed) Druid you know that some of the biggest stones at Stonehenge weigh around as much as six African elephants? Those Neolithic engineers were no numpties, but it’s safe to say getting them there was no plain sailing! (Get it? Because it ’s on a plain!

Why the stoney face? OK, enough now.)

Pythagoras wasn’t just The Triangle Guy. In the world of the ancient Greeks his investigations into maths and music caused a right old song and dance. We just hope this strikes the right note.

Choppy Waters: The Great Linguini is chained underwater at the bottom of an alligator tank! Can you unscramble the clues and set him free before he’s… erm… snapped up? Let’s hope so, for his sake. Here’s Brainfeeders.

The Great Linguini’s ice cream problem: You have just 20 minutes to sample 100 ice cream flavours and choose the best one.

Once a flavour has been sampled it must be chosen immediately or rejected forever.

How do you optimise your chances of making the right choice? Turns out there is maths for that. Who knew?

AQUILA news

It’s going to be OK! Our favourite Life Expert Kate Daniels is here to talk to us all about something everyone suffers from at one stage or another: ANXIETY. Honestly, I promise it IS going to be OK.

And the magic number is…

Goodwin-with-a-Y may have landed the job of his dreams but now he is in it up to his very neck. Can he interpret the workings of the Lucke Machine correctly? Is the nation bound for bloody war, or a jammy afternoon tea? Check out this brilliant brand new short story by Amy Sparkes, to find out.

Illustration: Tashi Reeve

Gimme s’more!

Harvey investigates a walkie-talkie-shaped building that set the world alight. No, not the world of architecture, or the world of walkie-talkies, I mean the actual world (small parts of it). On fire. No. It really did.

In William Shakespeare’s The Tempest, a powerful magician named Prospero survives on a tiny island using the magic he has learned from his books.

Prospero uses his magic to capture monsters and even raises the dead from their graves. (Spooky, ed.) Believe it or not, this extraordinary character is not a work of pure fiction; instead he is thought to have been inspired by a real person from the 16th century. A man whose life was filled with magic, mystery and mathematics.

John Dee was a mathematician, astrologer, astronomer and alchemist, who was born in London in 1527. He thought his research could solve all the mysteries of the universe, but could it?

Well, maybe some. Others, not so much.

John Dee started university in Cambridge at just 15 years old. He studied Greek philosophy and Arabic arithmetic and, after he graduated, he travelled around Europe collecting books, gazing at the stars and learning alongside other brilliant scientists such as the cartographer Gerardus Mercator.

John’s interests did not just include the sensible subjects we recognise today. He also studied occult subjects such as astrology, alchemy and natural magic.

Why would a sensible man like Dee be interested in magic? Ed At this time formal scientific study was still new. Our understanding of the universe was much more limited than it is today, and so magic and science were often seen as one and the same thing.

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People really couldn’t tell one from the other. Dee believed that numbers were the basis of all things and the source of all knowledge. Through numbers, he thought, humans could achieve divine power.

Dee eventually returned to England and settled into a comfortable life as a scholar and teacher. His skills were in demand, and he was able to use his knowledge of mathematics and the stars to advise a trading company about how to navigate new routes across the oceans.

THE QUEEN’S HEAD MAGICIAN Unfortunately though, his knowledge of astrology landed Dee in trouble with the law. In 1555 he cast the horoscopes of Queen Mary and her sister, Princess Elizabeth. This was a crime known as calculating, and it saw Dee imprisoned for three months. After Mary’s death, however, he became a scientific and medical advisor to Queen Elizabeth I, who believed in his magical powers. He even used his astrological skills to choose the date for her coronation!

Elizabeth didn’t just turn to Dee for magical, astronomical and mathematical advice, she also consulted him on medical matters. Although he had studied biological textbooks, John never actually qualified as a medical doctor.

Despite this, he was commonly referred to as Dr Dee. He was even sent on a journey to Germany to consult a famous physician about the health of the queen.

THE ORIGINAL OO7!

It has been suggested that Dee acted as a royal spy, who gathered intelligence for England as he travelled around Europe. Dee signed his letters to Queen Elizabeth I with the symbols ‘oo’ and ‘7’.

This mysterious sign may have influenced the 20th-century writer Ian Fleming when he came up with the codename for his best-known character James Bond.

We don’t know if Dee was a spy or not, but he did travel far and wide throughout his life to study new subjects and collect more books. His personal library is thought to have been one of the greatest private libraries in Tudor England. And he didn’t just collect books, he also had maps, globes and scientific instruments.

As Dee grew older he tried to explore the supernatural to help him understand things that mathematics could not explain. He started trying to contact angels through a crystal ball. It was through this work that he came to know a man named Edward Kelley. Kelley was a former forger who told Dee that he could talk to the angels for him. Dee was so convinced by Kelley’s skills that they soon began to work together regularly.

Nobody knows if Kelley tricked Dee, or if the two men worked together to mislead others, but they held séances to communicate with spirits and they travelled to Poland and Bohemia to try to win the support of their royal courts.

The two men parted ways in the late 1580s and Dee returned home to discover a very great loss.

When he went travelling with Kelley, Dee left his brother-in-law to look after his collection. When Dee returned home he discovered that his books had been sold or stolen. He was devastated by the loss of his treasured collection and, although he managed to recover some books, he never saw most of them again. Today, many of John Dee’s stolen books can be found in the library collections of institutions such as the Royal College of Physicians and the British Library. You can also visit some of the objects that John Dee used to communicate with spirits; the British Museum has his crystal ball and an Aztec relic made from volcanic glass called a ‘speculum’.

After the death of Queen Elizabeth I, the study of occult subjects was viewed

with increasing suspicion. The new king, James I, refused to support John Dee’s work and he died in poverty at the age of 81. His extraordinary life has been overshadowed by his relationship with Edward Kelley and his investigation into the spiritual world, but his brilliant mind led to advances in mathematics, medicine and navigation.

A STORM IN A DEE CUP

Words: Frances Durkin. Illustration: Kaley McKean

AN UNFORTUNATE FRIENDSHIP

(sorry, couldn’t resist, ed.)

William Shakespeare wrote The Tempest around a year after John Dee died. For people to believe that Prospero and Dee were connected is not surprising. These two men and their magical books still capture the imaginations of audiences and show that the world is always full of new ideas to explore. Even if some of them may be… a little odd.

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Maths isn’t just in textbooks and computers – it’s everywhere in the natural world too. Dogs, ravens, rats, and chimps all have ‘number sense’ – an inbuilt understanding of number systems. They can’t do long division or algebra, but Asian elephants count the members of their group to make sure no one has wandered off, and ants can do basic subtraction. Some lucky organisms can even do maths without thinking about it, building complex sequences and shapes into their bodies.

Sequences and spirals The Fibonacci sequence – a sequence in which each number is the sum of the two before it – appeared in Indian mathematics as early as 450 BCE, but it was first recorded in Europe in 1202 CE by an Italian mathematician called Leonardo de Pisa (or Fibonacci to his friends). He decided to use it to work out how quickly a population of rabbits could grow, explaining it something like this: Say you buy two baby rabbits, one male and one female, and put them in your garden. For the first month they’re too young to breed, so the number of pairs stays at one. After this, they literally start to breed like rabbits and produce a pair of babies at the end of the second month. Every month, the original pair and every new pair over a month old produce another pair of babies each, creating the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 and so on.

Of course, female rabbits don’t reach maturity until they are at least three months old. This example assumes that none of the rabbits get sick or are eaten, and that you have an enormous garden…

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The Fibonacci sequence pops up a lot in nature. Count the number of petals on a flower and often (but not always) it’s a number in the Fibonacci sequence, because these numbers seem to form patterns that help each petal get as much sunlight as possible. You can see it in trees too, as the trunk splits into more and more branches.

Divide a number in the sequence by the one before it and you’ll get an answer closer to 1.618 the further along the list you are. This special number (which actually has a lot more digits after its decimal place) is known as the Golden Ratio, and it’s represented by the Greek letter phi, which looks like this: φ. This ratio can be used to draw a spiral known as the Golden Spiral.

For every quarter turn the spiral gets wider by a factor of φ, creating an example of a type of curve called a logarithmic spiral.

Logarithmic spirals can be found in the seeds of a sunflower head, the arms of spiral galaxies, tropical cyclones and even the way a hawk flies towards its prey.

Platonic solids ‘The Platonic Solids’ sounds like the name of a 1970s pop group, but it’s actually another set of shapes found in nature. Named after the Greek philosopher Plato, they’re threedimensional shapes made up of identical polygons. These polygonal faces have equal length sides and equal angles, and the same number of faces meet at every vertex (or corner). You’ll definitely recognise the two simplest Platonic solids: a cube and a tetrahedron (or triangular pyramid). There are three other shapes that fit these very specific conditions; they’re called an octahedron, a dodecahedron and an icosahedron.

Fighting maths with maths No one has discovered a perfectly cuboid animal yet (oh, so SpongeBob

SquarePants doesn’t count then? Ed).

Not a cube. But we do know of

organisms that have bodies shaped like icosahedrons. Many microscopic viruses have shells made of identical triangular protein molecules, and an icosahedron is the most efficient way of arranging them. Building their bodies from lots of identical pieces means that viruses only need to have very simple DNA, allowing them to reproduce easily inside the cells of other organisms.

Viruses can use their simple bodies to quickly grow in number and spread from person to person. Left to their own devices they can wreak havoc, so when an infection starts to move through a population, it’s time to fight maths with maths! By combining mathematical theory with knowledge about the area, its level of medical care, immunity among the population and information about the virus, experts can predict how the illness will spread and work out how to use resources most effectively.

Was maths invented by humans, or was it already here and just waiting to be discovered? It’s a question we will probably never get a complete answer to. We do know this, however: once you discover the magic of numbers, you’ll begin to see it everywhere!

Words: Victoria Williams. Illustration: Stacey Thomas

Probably the best-known logarithmic spiral in nature is the shell of the nautilus (Nautilus pompilius). A nautilus is a cephalopod, related to the squid and the octopus. While its relatives brave the dangers of the sea without armour, the nautilus keeps its soft body tucked inside a smooth curving shell, sticking its bird-like beak out to feed and propelling itself backwards through the sea by shooting water out of its body. The logarithmic shell keeps the nautilus safe from predators but not from the pressure of the ocean’s depths; roam more than 785 metres from the surface and the animal implodes (what a way to go, ed).

FUN FACT TRUMPET A newly hatched nautilus has a shell with four compartments and this increases to 38 or more in fully grown adults!

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We all ‘know’ Pythagoras, right? The ancient Greek dude who may or may not have existed around 500 BCE. He’s the one with THAT theorem about right-angled triangles.

Oh yeah, I know the one: In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sideszzzzzz.

Ed.

The blacksmith told him that the hammers he used were different weights – six pounds, eight pounds, nine pounds and twelve pounds. Each one created its own sound when it struck the iron. Pythagoras realised that the perfect ratios (half as heavy, and two-thirds as heavy) caused a perfect change in pitch.

So does that mean musical notes have numbers connected to them? Ed.

Wake up!

But did you know that as well as being a mathematician, Pythagoras was also a philosopher who believed that the universe was in harmony and that maths could prove it?

Yes! Pythagoras had discovered that pleasant-sounding music depends on maths! Every time you go up an octave (8 notes) the hertz (frequency of sound waves per second) doubles!

Magic!

Oh AND he was a religious leader with loads of followers who lived their lives according to numerology and other weird and wonderful rules?

WHY THE OBSESSION WITH NUMBERS? Pythagoras loved maths because it contains steady patterns – certain things ALWAYS happen. He also thought that, because it never changed, numbers could explain the truths of the world and the universe. Rather unexpectedly, he found out it was the secret to music too.

HAMMERING OUT A TUNE One day, Pythagoras was walking past a metal-making forge just as the blacksmith was hammering the metal on his anvil. Pythagoras noticed the ringing. Some hammers rang with a high note, some with a pleasant musical pitch higher, and yet another was a pitch lower. Some were harmonious, others less so. So, being curious, he went inside to find out why this was happening.

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Indeed!

You can do one of Pythagoras’s experiments: 1 Take four identical glasses: • Pour 200ml water into the first • Pour 100ml (half the previous) into the second • Pour 70ml (nearly 2/3 of the first) into the third • And pour only a little into the fourth so that it’s nearly empty.

2 Tap the outside of each glass (gently) with the edge of a metal spoon. You will hear them each ring out with a different note.

They will sound pleasingly similar, but higher in pitch.

Until the nearly empty one – it usually sounds very poor – it’s not filled to a mathematical ratio of the others! Maths makes music!

FUN FACT TRUMPET Scholars cannot agree whether or not a man called Pythagoras ever existed. Not much is known about his life, because most records were written hundreds of years after he was meant to have lived.

This uncertainty is known as the Pythagorean question.

MEASURING MUSIC Pythagoras then looked at a lyre (an ancient Greek stringed instrument similar to a harp).

He noticed that two strings with the same length, thickness and tension (tightness) sound the same.

He realised that making the lengths of strings on the lyre proportions of each other, produced perfect, neat mathematical intervals or gaps between notes.

So, if one string is exactly twice the length of another, the short string note is much higher, but they sound good together. This is because they are an octave apart (the same note).

If one string is two-thirds the length of the other, they still sound good, although one is a higher pitch.

Musicians now call this gap a Perfect Fifth.

Okay, so what about maths producing the harmony of the universe?

THE HARMONY OF THE SPHERES Pythagoras realised musical harmony was mathematical. So, he said, the universe must be perfectly balanced or the stars and planets wouldn’t spin in their predictable circles.

Then he suggested (wrongly) that the Sun, Moon and planets and ‘special stars’ must emit their own music, each in a perfect pattern or interval with each other. He called this the ‘harmony’ of the spheres.

What does ‘the square’ or ‘squared’ mean?

4 In maths, if you multiply that 4 cm edge by itself you get 16 (4 x 4 = 16), this means squared.

It’s as if you had a piece of chocolate which was 4 blocks wide and 4 long – you would have a square made up of 16 blocks of chocolate! (Mmmm

chocolate, ed.)

5 The 3 cm squared is multiplied by itself (3 x 3) = 9.

6 Add the 16 and 9 together = 25.

And this is the same as the hypotenuse multiplied by itself (5 x 5 = 25!).

Pythagoras said that he could hear it but his fellow humans couldn’t, because we are too used to it, just like we don’t feel the clothes on our bodies unless we think about it.

This wasn’t science. Or maths. This was philosophy; it was about understanding that mankind has an intimate, personal relationship with the universe and that when maths is in harmony, we find it beautiful.

(And that means that I also have 50 blocks of chocolate. Winner!)

Words: Gillian Hovell. Illustration: Nolan Pelletier

FUN FACT TRUMPET Pythagoreans (Pythagoras’ followers) lived their lives according to certain rules, such as taking their right shoe off first (but washing their left foot first) and not picking up what has fallen (had they not heard of the 5-second rule, ed?).

And let’s not forget that theorem…. try it for yourself!

1 Measure and mark 4 cm along the bottom of a piece of A4 paper from the bottom (right-angled) corner.

2 From the same corner, measure and mark up 3 cm.

3 Draw a line to join these two points.

Measure the line – it will be 5 cm.

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Stonehenge is a mysterious prehistoric site on Salisbury Plain, Wiltshire, in the south-west of England. Some people think it was built by giants, others say it was constructed by druids, and others say aliens!

Whoever the builders were, they were certainly very clever.

FUN FACT TRUMPET The heaviest sarsen stone weighs in at nearly 36 tonnes – that’s about the same as two double-decker buses loaded with passengers! Imagine dropping that on your toe! (Ouch, ed.)

What makes this pile of rocks so special? Ed Archaeologists believe that Stonehenge was built around 4,500 years ago by people from across the British Isles who were really good at maths, engineering and astronomy. Parts of it have fallen down or are missing, but Stonehenge is still so amazing that every year more than a million people come to see it from all over the world!

GOING ROUND IN CIRCLES The most famous part of Stonehenge, and the bit you’ve probably seen photos of, is the stone circle.

But outside of it is another bigger circle made from a ditch and chalk bank – this is the actual henge and it was built about 5,000 years ago. It is about 100 metres in diameter. The stone circle inside the ditch and bank was built from two different types of stone about 500 years later. The biggest stones in the circle are made of a very hard rock called sarsen. The tallest of these from base to tip is 8.71 m, but 2.13 m is buried underground. We don’t know exactly where the big sarsen stones were brought from, but it’s likely they are from somewhere like the Marlborough Downs, which is 25 km (15 miles) away from Stonehenge. This is at least a day’s walk – and that’s without carrying 75 massive stones!

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HEAVE HO!

The smaller stones are called bluestones and they are a type of rock that does actually look blue, especially when it’s raining. Some people say they have musical properties! Even though these are the smaller stones they’re still over 2 metres high – probably bigger than most adults you know! These rocks (originally there were 80 in total) were brought all the way from the Preseli Hills in south-west Wales, 220 km (136 miles) away. These stones must have been pretty special to have been brought up and down hills, round corners and even over water and THEN they had to be stood upright!

No one had thought about inventing the wheel yet, let alone engines, so we think people would have put the stones onto wooden sledges and draaagggged these along track ways, or used rollers to move them. They may even have used boats to bring them along the coast from Wales and up the River Avon to Salisbury Plain.

another side sloping, and then the stones would have been pulled upright into position using ropes made out of plants!

Then we think they built wooden scaffolding and used levers to raise the heavy, horizontal stones into position on top of the upright stones.

Wow, very ingenious, but it seems like a lot of effort. What was the point? Ed Well...we’re not really sure!

LOOK TO THE SKIES We do know it would have taken a lot of people hundreds of years to complete. It is likely people took part for religious reasons. One of the cleverest things about Stonehenge is that it was built to line up with the movements of the Sun. Although they are in a circle, the stones are arranged along a line running from north-east to south-west. If you stand in

the centre at sunrise on the longest day of the year (midsummer), you can watch the Sun rise over the Heel Stone. On the shortest day (midwinter) you can watch the Sun set in the opposite direction, between the two stones of the tallest trilithon (two vertical stones with one horizontal stone lying across the top) at the head of the inner horseshoe. That’s pretty clever, isn’t it?

We will never know for certain what Neolithic people thought about the Sun and the skies, but we think they probably gathered here to mark the changing of the seasons and the turning year. Light meant life for prehistoric people. They relied on the Sun to grow crops and raise healthy animals. If they saw the Sun as a god, maybe they gathered here in the middle of the winter to pray for the return of light?

What do you think?

The people who built Stonehenge lived in what is known as the Neolithic period; this is the last part of the Stone Age. Metals had not yet been engineered, so people made their tools from stone, wood, bone and antler. They used big, round stones called hammerstones to work the sarsens and bluestones into rectangular shapes, and to make special joints to join the stones together. These joints are called mortise and tenon joints. They were used so that the horizontal stones (some of which weigh 8 tonnes – heavier than an African elephant!) could sit across the top of the standing stones without sliding off and, you know… smooshing someone important.

To stand the stones upright, holes would have been dug in the ground. One side of the hole would have been straight and

BUILD A BISCUIT HENGE

Why not have a go at making Stonehenge...out of biscuits! Shortbread biscuits make excellent sarsen stones and pink wafers will balance well to make the linteled structures. It might be fun to use a small tray and collect stones, moss, grass and soil from outdoors to create a more realistic version. Try shining a torch across your monument to show solstice sunrise and sunset!

Words: Susan Greaney. Illustration: Sophie Bryant-Funnell

STONE ME, THAT’S CLEVER!

To make a really accurate model, you can use the measurements in the diagram above to calculate an appropriate scale. A scale of 1/10 is best for something small enough to fit on a table!

Many thanks to Susan Greaney and English Heritage who do great work conserving Stonehenge so people can continue visiting it for thousands of years to come.

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It is the interval at The Great Linguini’s magic show and, naturally, you are first in the queue for an ice cream. You have 20 minutes to choose a flavour, and you can try 5 per minute. That’s a maximum of 100 ice cream flavours before the interval ends.

The flavours are being introduced one by one at random. You have no idea what is coming next, but you’ve decided to give each one a score out of 100 as you go.

THERE ARE TWO RULES: Rule 1: When you taste an ice cream flavour you must either choose it or reject it. Once you have rejected a flavour you can’t go back. That’s it, it’s gone forever.

Rule 2: If you have not chosen a flavour by the time you get to the 100th , you are stuck with that one.

Even if it is fish guts. Soz about that.

Your mission is to choose the best possible flavour of ice cream, but how?

Can you come up with a plan to guarantee the finest interval snack?

There is a mathematical theory that means you can select the optimal ice cream flavour more than one third of the time.

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WHY DOES IT WORK? Sample LOTS If you sample too many flavours before choosing, you’re likely to end up going straight past the best flavour. Then you’ll search in vain for a better flavour that doesn’t exist, before getting stuck with #100.

Sample just a few If you only sample a couple of flavours then you might not catch any of the really scrumtastic ones in the sample. If the best flavour in the sample was, for example, only the 30th best overall, then you’ll choose the first flavour you come across that’s better than 30th best. That’s not likely to be the VERY best one.

Goldilocks spot What we have to do is find the Goldilocks spot – not too many, not too few – to optimise our chances. You can see what’s happening in the graph:

The Magic Number e Behind the scenes there are some tricky things going on, and they all come back to a truly magical number: e. The number e (= 2.718…) crops up in maths all over the place. Here, it gives us both the chance of getting the best flavour (1/e ~ 0.37), and it tells us the optimal sample size by using the formula n/e, where n is the number of flavours in total.

First, taste the first 37, score them out of 100 and dismiss them.

This will really annoy your server and everyone else in the queue, but screw your courage to the sticking place* (or the licking place, it is ice cream after all, ed).

Then, keep tasting and ranking. When you find a flavour that beats everything in your sample: choose it. You will find that, 37 per cent of the time, you end up with the best possible result.

If you are brave enough to try this in real life, then you’re about 37 per cent braver than any of us. Ed.

*That’s from ‘Macbeth’ Macbeth’ – oops, I mean The Scottish Play – we are in a theatre!

This works for really big numbers, but there are also variations for smaller numbers. If there are just THREE flavours then: A sample the first one (and don’t choose it!)

B taste the second one. If it’s better than the first: choose it! If not: continue…

C and choose the third one without trying it.

You’ll get the best flavour 50 per cent of the time.

That’s better odds than if you had guessed at random, and it’s quicker than trying all three before deciding! (Plus, if the first one really was the best then you’ll have an excuse to go back another time…)

Words: Freya Hardy and Sylvia Morris. Illustration: Ed J. Brown

TRY IT OUT:

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Words: Freya Hardy, Benita Estevez and Liam Findlay. Illustration: Sophie Bryant-Funnell

Gaze back in the mists of time for a minute with me.

Imagine it’s about 4,000 years ago.

You are standing on the banks of the Yellow River in China, stroking your long beard and trying to figure out how to stop the regular flooding that repeatedly devastates the villages here.

Your name is Fu Xi, also known as Fu Hsi, one of the mythical emperors of China, and you’re in a bit of a pickle.

Your people are looking to you for answers. The good thing is that, according to legends of your greatness, you’re extremely wise and practical, so the answer will no doubt be along soon.

Whoa! What’s this? A tortoise shell (sometimes a dragon horse in the old stories). That’s a good omen, and it has a unique pattern on its back. Later known as the Lo Shu Grid, this pattern will come to form the basis of the Chinese practice of numerology. This grid will be used to calculate everything, from where to build your office complex to what date to get married on.

Hang on, that ’s all well and good but did Fu Xi ever figure out the flooding problem? Ed It’s difficult to find out much about Fu Xi, but the stories say he did stop the flooding by using canals or dams to divert the water.

OK. Phew.

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Take a closer look at this grid, what do you notice?

Exactly, you mathematical whizz, you. Every line –­ horizontal, vertical and diagonal, adds up to 15 – roughly the number of days between a new and a full Moon. As many celebrations and festivals are held around the Moon’s cycle, the 15 days act as a useful marker for plotting its progress.

No way. I don’t believe you, ed Why else would the 08/2008 Beijing Olympics start at 8.08pm? I’ll give you a clue, it wasn’t because the head flame carrier needed a pee. The numbers eight, six and nine are considered auspicious (that means promising). The number six is lucky because it sounds similar to words like ‘slick’ and ‘smooth’ in Cantonese, as well as the phrase ‘good fortune’ in Mandarin. Eight sounds similar to the word for wealth, and nine is historically associated with the Emperor of China, a powerful and imposing chap.

Are there any inauspicious numbers?

numbered 4 or 44 tend to be occupied by foreigners.

Any science to back this up?

No. None whatsoever. But the ancient practice of Chinese Numerology has undoubtedly shaped the world, regardless.

And in any case, science or not, numerology in all its forms is a system of beliefs that many people still hold on to today.

Whether you feel for the vibration of certain numbers to make your choices, or look to the stars for your answers, you’re following a system that, while completely unscientific, is a lot of fun and has its roots in traditions that are thousands of years old.

The numbers 4 and 7 are both considered

OK. What do we do with this information? unlucky. Seven represents the month of As well as representing a number, each square also holds other meanings connected with family, wealth, work and so on. People who believe in numerology think these can be used to calculate what is missing in areas of your life.

There are plenty of modern takes on this ancient belief system, but the grid remains prominent in Chinese culture, and so do the beliefs built around it.

July, which is also known as the ghost month. The Chinese word for ‘four’ sounds a lot like the word for ‘death’ in Mandarin and Cantonese dialects. As a result, lots of apartment buildings in China don’t have a fourth floor. Instead they have a 3a and a 3b, followed by a 5. Apartments

Fun Fact Trumpet A lot of Chinese superstitions come from homophones (words that are pronounced the same but have a different meaning). It is bad luck, for example, to give someone in China a clock as a present, as the term for this is a homophone for the term to attend someone’s funeral!

Words: Caroline Pattenden. Illustration: Jasmine Floyd

Lo Shu Grid

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PREPARE THE STAGE 1 Cut the bottom flaps off the box. Close up any

From the late 1800s to the middle of the 20th century, escapology, magic and illusion were super popular theatrical art forms in Europe and North America. Stars like Harry Houdini toured alongside acrobats, trained animals, jugglers and comedians in vaudevillian variety shows. Houdini had a wide range of skills for escaping deadly situations, but he also had some nifty equipment and conjuring techniques to amaze and astound his audience.

Fancy making your own one-person variety show? Step right up and give it a go!

YOU WILL NEED: – 1 large cardboard box* – Ruler – Scissors or scalpel – Strong tape – Wrapping or wallpaper – Plain black paper – Scrap paper – Scrap card – Adhesive foam pads (optional) – Chalk pens, paints or felt tips – Glue stick – Fabric – approx 40 x 50 cm – Printer (optional) – Twisted rope cord – Compass (optional) – Double-sided tape (optional) – Helpful adult *A larger box can be cut to size

other flaps with strong tape.

2 To make the armholes, measure 15 cm up from the open bottom of the box. This is the bottom of the armhole. Draw an oval about 20 cm x 12 cm.

Repeat on the opposite side.

3 Carefully cut out the ovals (adults can help here) and cover the edges with tape.

4 Wrap the box in wrapping or wallpaper.

5 Cut out the armholes and secure with tape.

6 Leaving a 1 cm border around the edges, attach a rectangle of plain black paper to the front of the box.

MAKE THE TOP PIECE 7 Fold some scrap paper in half. This will make your design symmetrical. Draw a fancy-shmancy design like the one pictured here, and cut it out.

8 Now unfold. This is your template.

9 Draw around the template onto card, then onto wrapping or wallpaper. Leave 2 cm extra on the wrapping/wallpaper. You can snip into this to create a fringe. Stick the pretty paper to the card.

10 Draw the letters to make the word THEATRE. You can search circus fonts and print them out if you prefer.

11 Draw around your letters onto thin card. Cut out and colour in.

12 Cut out a rectangle as a background mount for your letters. Cover it with coloured paper.

13 Attach the letters using sticky foam craft pads, double-sided tape or glue stick. Stick to the ornate top piece.

14 Cut your fabric into two pieces 20 cm x length of your box. Turn over the top piece and tape the fabric in place.

WHO’S THE STAR OF THE SHOW? 15 Draw your star (you can print out a picture if you prefer). We chose Houdini for a vaudeville vibe but you can make up a magician of your own.

16 Ask an adult to cut out a gap for your eyes in the image and the box. Pay attention to the placement if you want to look out through the gap! Attach the image to your box.

FINISHING TOUCHES 17 Attach the decorative top to the top of the box.

18 Make two holes half way down on each side of the front panel. Thread the twisted cord through the holes and use to tie the curtains.

19 Add decorations wherever you please.

20 Climb in, break a leg and become the star of the show (just make sure you can get out again, ed).

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Words and photographs: Let’s Make Art, Bristol. Visit their Instagram @letsmakeartuk

Plain black paper

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Did you hear about the building in London that could fry eggs?

You’re cracking me up! Ed No. It’s not a joke. Honestly!

The curved shape of the building was causing a problem; only for part of the year, for a few hours a day and only if it was sunny… the right combination of conditions doesn’t happen in London quite as often as you might think!

In 2013, reports emerged of mysteriously melting car trims, a scorched doormat and even a bicycle seat catching fire! The culprit was very quickly identified. It was 20 Fenchurch Street!

As the sun reached a certain point in the sky, it would shine onto the glasscovered building and be reflected onto the street below. So far, so normal.

However, the bowl-shaped paraboloid, curved surface of the building was focusing the Sun’s energy and causing things to heat up. The owners of the building had to add sunshades to prevent the reflections from causing future problems.

The building at 20 Fenchurch Street is known as The Walkie Talkie because of its distinctive shape. It’s a clever design because although it has a very small footprint at ground level, the floors get bigger as you go upwards. However, this 37-storey building quickly attracted media attention for a totally different reason. On sunny days, people were seen frying eggs on the ground in front of it. It became known as the Fry Scraper and the Walkie Scorchie!

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You might not have realised it, but you’ve experienced this paraboloid effect before. Satellite dishes work by collecting lots of weak signals and focusing them together to create a stronger signal. The dish has a paraboloid-shaped surface, which collects microwave signals from satellites and focuses them back onto the feedhorn, or collector – the structure on the end of the arm in the middle of the dish.

FUN FACT TRUMPET In sport it is important to be able to follow the path of a parabolic trajectory. For example, when you kick a football or throw a cricket ball it will not travel in a straight line. Instead it will follow a parabolic path (spin and air resistance may distort the shape) – and knowing where they are going to end up could mean the difference between winning and losing a match!

the angle of incidence is equal to the angle of reflection. If the dish is the right shape, no matter where the signal hits the dish, it will be reflected to the same point; the focus (F). This is where the collector is placed.

This method has also been used to collect sound from great distances.

Between the end of World War I in 1918 and the start of World War II in 1939, a network of acoustic mirrors were built on the south coast of England. A microphone placed at the focus detected the sound of aircraft approaching. They were replaced by the radar system, which sends and receives radio waves in a similar shaped dish. Any aircraft hit by a radio wave would reflect it back to the collecting dish. Radar makes it possible to work out the number, size and location of aircraft, so it quickly became the preferred detection method.

The diagram shows a slice through a paraboloid. This is called a parabola. To understand how the physics works, you have to think of the curve as being made of lots of tiny straight bits joined together. This means that any point on the curve is a tiny bit of straight line (shown by the thin black line). Follow line A on the diagram: it hits the dish with an angle of incidence of α: the angle between line A and the dish (blue). It is then reflected with angle α: the angle between the dish and the reflected line.

The paraboloid collecting dish works because the signal that reaches it bounces off the surface at the same angle it arrived at. In scientific terms,

EXPERIMENT! You will need: – Marshmallows – Biscuits (chocolate digestives are best) – Wooden barbecue skewer (or metal one with hand protection) – A curved balti dish, wok or bowl – Aluminium cooking foil – A sunny day (!)

– An adult to supervise you – this could catch fire! If where you live has fire bans in place, please adhere to them correctly before trying this experiment.

1 Cover the inside of your pan or bowl in aluminium foil. Try to make it as smooth as possible, with the shiny side facing towards you. This is your solar cooker.

2 Take your solar cooker outdoors and

4 Wait and watch. Can you see the

place it in the sunshine. Angle it towards the Sun for best effect.

marshmallow start to brown and caramelise? When it’s ready, sandwich it between two chocolate digestives (chocolate sides inwards!) and enjoy your solar-powered snack!

3 Put your marshmallow on the skewer and hold it inside the middle of the solar cooker, about 5 cm from the base. Adjust your marshmallow until it is really brightly lit from all sides. This is the focal point. Keep your fingers well out of the way of this bright point – it should get very hot!

Words: Sarah Bearchell. Illustration: Ed J. Brown

Turn a parabola into a SOLAR COOKER to make s’mores!

Make sure you keep your solar cooker out of sunlight when not in use – you don’t want to cause any accidental fires!

Focal point

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Bear with me here. Imagine right now that you are a wild animal (well, you did say to ‘bear’ with you! Ed). I bet there are a few things you would be sure to do every day, the most important things being finding food and water, and keeping yourself (and maybe your family) safe.

BEAR NECESSITIES So here’s another question. Do you think wild animals get anxious? I think they do, especially the small ones! In fact, I reckon that lots of animals, besides us, get anxious, don’t you? I know I would, if other animals wanted to eat me!

Yep, lots of different animals seem to get anxious, but human anxiety is often different from animal anxiety, because human anxiety tends to be more complex.

Let me give you a couple of examples: If a bear has a problem, e.g. ‘I am thirsty’, it will find some water, its problem is solved. Simple! If a human has a problem, e.g. ‘What if I fail my maths exam?’, they study really hard, get anxious when they don’t understand or remember things, worry some more at night, talk to their friends or family/carers about it, worry some more, think they are a failure etc.

You see? Complicated!

DECISIONS DECISIONS All animals make lots of decisions every day. The big difference with us is that these days we tend to worry most about things in the future. Researchers call this delayed return. It means that lots of our daily choices aren’t made to solve immediate problems. The problem with this ‘delayed return’ society is that we are

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not really designed well for this stuff (in the distant past we only had simple needs like the bear), so it makes us all feel super-stressed and anxious.

We feel anxious because we are not using our anxiety to do the job it was designed to do, i.e. hunt for food and water. We haven’t had time to evolve because our modern lives have changed so quickly – essentially, our brains can’t keep up. Anxiety is a totally natural feeling for us all, but knowing this doesn’t always make it easier to deal with.

I’m sure you have experienced feeling anxious or worried – maybe when you have had to do something that tests you in some way? Normally, in these situations, we feel worry or anxiety and then, after a short while, we feel calmer.

This is how it is supposed to work but sometimes we can find it hard to calm down easily and can then feel anxious a lot of the time. That is not good for our emotional health.

SO WHAT DO WE DO ABOUT IT? According to YoungMinds charity, if your anxiety gets worse or the feelings last longer than they should, it could be time to get some support. They list some of the possible symptoms on their website1. Having some or all of these anxious feelings doesn’t necessarily mean you have high levels of anxiety, but if they are affecting your day-to-day life, it is important to tell someone you trust about how you are feeling and perhaps talk to a doctor. With proper help and support it has been proved that even strong anxiety can be treated and you can get better.

There are some simple things we can do for ourselves that have been shown to really help lower anxiety. Let’s look at some of these:

1 Breathing calmly When you feel anxious, concentrate on your breathing. Relax and slow it down, imagine clear air going deep into your lungs and then let it out slowly. Breathe in calmness, breathe out anxiety.

2 Talking about how you feel Find an adult that you trust to talk to when you feel anxious.

Maybe your parents, carers or other close family members or a doctor, counsellor or therapist. (Don’t forget your friends too!)

3 Daily relaxation Every evening try reading an inspiring book or do a meditation2, listen to calming music or maybe use an app to help manage anxiety. (The Mental Health First Aid (MHFA) recommend ‘Positive Penguins’3 as an example.)

4 Physical exercise Get out for a walk as often as you can with a parent or carer. Join a sports club at school or in your local area. Dance around the kitchen or run around in a park or garden together.

5 Getting enough sleep Children need 9 to 11 hours sleep every day, so this would be a good place to start. Put all online devices away an hour before you sleep (the light from them can keep us awake and our brains working), write down all the things that are worrying you in a special ‘feelings’ book and/or write a list of all the things you are grateful for.

6 Finding moments in your day to be mindful Notice the weather and how it feels on your skin, a friend’s smile, the sounds as you walk home from school etc, be aware of everything and try and be in ‘the moment’ as often as you can every day.

We can’t stop being human, we can’t stop getting anxious, but, with the help of others, we can learn to find ways through it.

1 https://youngminds.org.uk/find-help/conditions/anxiety 2 Relax Kids by Marneta Viegas is one of our favourite meditation books for children

understand their feelings and challenge negative thinking.

This feature was written by Kate Daniels who runs PSHE Education (psheeducation.co.uk) and spends her time educating children, their teachers and parents on how to deal with life in the modern world. She loves teaching and writing and can’t think of anything she’d rather be doing! Before this Kate was a primary school teacher for over 10 years. She is mum to two far-too-grown-up-kids and lives in the East Sussex countryside with her old, hairy horse and a rather squeaky cat!

Words: Kate Daniels. Illustration: Stacey Thomas

3 positivepenguins.com is an app aimed at 8–12-year-olds. The app helps children to

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In an 8-shaped building within the castle

It was a simple, but careless error: they

Distracted by the violent confiscation of

walls, lived Nineteen Numerologists.

didn’t realise Goodwin spelt his name

his bag by a miffed greengrocer,

with a ‘Y’ not an ‘I’. This turned his

Goodwyn was unaware that the apples

Name-Number from a reliable 4… to a

had rolled through the market, startling

disastrous 6!

five hens, which flapped straight into the path of two horses, ridden by the Queen

Goodwyn-with-a-Y was hired in great

of the Westlands’ heralds. The horses

haste. The Chief Numerologist had

reared and the heralds exchanged grim

retired unexpectedly that morning to

looks, confirming their suspicions of

take up experimental goat farming,

ill-fortune about this visit. (This had

leaving the Numerologists understaffed

nothing to do with numbers, and

(and slightly baffled). Even worse,

everything to do with the punch-ups that

minutes after his departure, there came

tended to occur when the queen and king

an announcement: the Queen of the

met for Important and Unpleasant

Westlands – the dodgiest kind of

Discussions).

7-Name-Numbered person – was coming for an Important and Unpleasant

Whilst the heralds wondered if they’d

Discussion.

brought enough sandwiches for the duration of a punch-up, the bagless

So, after a hasty recruitment campaign

Goodwyn arrived at the 8-shaped

lasting precisely 17 minutes (17 minutes

building and knocked on the door four

x 60 seconds = 1020 seconds; 1+0+2+0

times.

Their job? To undertake Extremely

= 3, and 3 is a good, solid number),

Complicated and Clever Calculations and

young Goodwyn was invited to join

Now, 4 was good. A strong, solid number.

counsel the King of the Heartlands.

them.

The Deputy Chief Numerologist opened the door. He would never open it for a

But no one checked the spelling of his

2- or a 5-knock knocker. He did not know

name. They were too busy making

that Goodwyn’s hand had been raised for

With their Extremely Complicated and

Extremely Complicated and Clever

a fifth knock, but he’d been distracted

Clever Calculations, they advised His

Calculations, and burning incense to help

by the noise of five hens flapping

Majesty which date to fight battles, which

them concentrate (it disguised the smell

overhead.

queen to choose (avoid dodgy 7-Name-

of the Fourth Numerologist’s feet).

Their worth? Incalculable (annoyingly).

So, fatefully, across the threshold the

Numbered ones), and even which food to serve the warmongering King of the

So Goodwyn entered the castle gates,

6-Name-Numbered, 5-knock-knocking

Northlands when he dropped in for tea.

bringing only a bag dangling from a stick

Goodwyn stepped, his nose twitching at

(Thanks to their calculations, wars were

and a sense of optimism – both of which

the heady, incense-filled air. Goodwyn

won, the king was happily married, and

would soon vanish.

had always admired the Nineteen Numerologists and couldn’t believe that

the chocolate sponge was removed from Through the bustling market he walked.

today, he would join them. (Because it was

He stepped sideways to avoid a pile of

indeed unbelievable: there was absolutely

Their mistake? Hiring a young apprentice

manure and his stick-bag dislodged some

no way this boy should have been allowed

named Goodwin.

apples on a cart.

anywhere near the building.)

the oven before burning).

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5 + 15 + 18 + 23 + 6 = 67.

And whilst an egg seller argued with the

crucial element to the Calculations: a bit

heralds about who exactly was

of Lucke. Obeying instructions, he rolled

responsible for her missing chickens,

the hexagonal box in its frame three

the Queen of the Westlands trotted ever

times (a good, solid number). Inside, one

closer towards the castle walls.

hundred little brass balls clattered and

Four – good, dependable. Four advised

clashed. With trembling hands, the boy

peaceful negotiation.

Fate was in the balance. Time was

undid the latch and extracted five balls

running out. The Numerologists needed

– each engraved with a number.

6 + 7 = 13 and 1 + 3 = 4.

But which was truly the final component

to be sure whether to advise His Majesty

of the Extremely Complicated and Clever

to prepare for battle, or pop a sponge in

Calculations? Which number was right?

the oven. This involved 693 Extremely

6 or 9? War or peace? Soldiers or cake?

Complicated and Clever Calculations.

Goodwyn stood awkwardly, as the

The Numerologists completed their

Numerologists chalked on slates, slid

tasks and turned to Goodwyn.

balls on giant abacuses and worked out Out of time.

how many eggs were needed for a giant celebration cake if they advised peace.

Heart thumping, he made his choice.

And whilst the angry greengrocer in the market waved a stick-bag at the heralds

* * *

and pointed at the 8-shaped building, Goodwyn rubbed his itchy, incense-

On the hills, oblivious of the chaos,

blasted nose and was shown a large,

ex-Chief Numerologist Tom

long hexagonal-shaped wooden box:

5, 15, 6, 23, 18.

the Lucke Machine, named after the

Bowler‑Lucke sat contentedly. He’d thought of explaining that the Lucke

ex-Chief Numerologist himself, who

As the Numerologists scribbled,

Machine cheated, that balls were

created it: Tom Bowler-Lucke. It was

counted and panicked, the fate of the

weighted to produce certain

mounted within a frame so it could

kingdoms was literally in Goodwyn’s

calculations, but his colleagues were

rotate and had a little latched door in

hands. But fate is cruel, and at that very

convinced of its merits. And they all so

its middle.

point, Goodwyn did something that no

enjoyed the cake...

Extremely Complicated and Clever Calculation could cleverly calculate.

were not interested in chickens, apples or stick-bags and just wanted to announce the Queen of the Westlands’ imminent arrival, preferably BEFORE she arrived, the Deputy Chief Numerologist hastily explained to Goodwyn his task.

As Goodwyn listened carefully, the Queen of the Westlands entered the castle gate. And as the Queen rolled her eyes at her useless heralds surrounded by angry market folk, the Deputy Chief Numerologist ran for the castle, to advise that under NO circumstances should His Majesty wear the woolly overgarment depicting the red-nosed stag, which his mother had gifted him

He could tolerate the incense no longer.

He sneezed.

The balls tumbled out of his hands, scattering everywhere. Goodwyn gasped in horror, and hastily retrieved them.

15, 5, 23, 18, 9…

Distant bells announced the news: the castle ovens were alight. Tom smiled.

Because for all the Numerologists’ intelligence, no one had thought to check the spelling of his name. There was no ‘E’ in Lucke. He was a dodgy 7-Name-Numbered person… albeit a nice one.

Well, good Lucke to them. Tom watched his goats happily and ate some retirement cake.

Or was it 6?

Goodwyn tried to remain calm.

5 + 15 + 18 +23 + 9 = 70.

last Christmas.

7 + 0 = 7.

Thus, poor Goodwyn was left

Seven – self-serving, deceptive. Seven

unsupervised to provide the final,

advised war.

Illustration: Nolan Pelletier

And whilst the heralds explained they

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