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Pythagoras

Over 2000 years ago there was an amazing discovery about triangles:
When a triangle has a right angle (90°) ...... and squares are made on each of the three sides, ...... then the biggest square has the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

Note:

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Let's check if the areas are the same:

3 ² + 4 ² = 5 ²

Calculating this becomes:

9 + 16 = 25
It works ... like Magic!

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a ² + b ² = c ²

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

Read Builder's Mathematics to see practical uses for this.

Also read about Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle.

Example: What is the diagonal distance across a square of size 1?

It works the other way around, too: when the three sides of a triangle make , then the triangle is right angled.

Example: Does this triangle have a Right Angle?

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Yes, it does!

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

Draw a right angled triangle on the paper, leaving plenty of space.

Draw a square along the hypotenuse (the longest side)

Draw the same sized square on the other side of the hypotenuse

Draw lines as shown on the animation, like this:

Cut out the shapes

Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

becomes

We also have a proof by adding up the areas.

Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.

511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361

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Pythagoras Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... ... then the biggest square has the exact same area as the other two squares put together!

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From bathroom floors to honeycombs or even groups of cells, tilings surround us. These patterns cover a space without overlapping or leaving any gaps. Like a rug filled with diamond shapes, where each section looks the same as the one next to it, every tiling ever recorded has eventually repeated itself-until now.

After decades of searching for what mathematicians call an "einstein tile"-an elusive shape that would never repeat-researchers say they have finally identified one. The 13-sided figure is the first that can fill an infinite surface with a pattern that is always original.

Repeating patterns have translational symmetry, meaning you can shift one part of the pattern and it will overlap perfectly with another part, without being rotated or reflected. The shape described in a new paper does not have translational symmetry-each section of its tiling looks different from every part that comes before it.

Sarah Hart, a mathematician at Birkbeck, University of London, who didn't contribute to the finding, tells New Scientist'sMatthew Sparkes that she had thought finding an "einstein" (named for the German words for "one stone," or one tile) could not be done. "There are infinitely many possible candidate tiles, and even the existence of a solution feels quite counterintuitive," she says to the publication.

"Everybody is astonished and is delighted, both," Marjorie Senechal, a mathematician at Smith College who did not participate in the research, tells Science News' Emily Conover. "It wasn't even clear that such a thing could exist."

David Smith, a retired printing technician and nonprofessional mathematician, was the first to come up with the shape that could be a solution to the long-standing "einstein problem." He shared his ideas with scientists who took on the challenge of trying to mathematically prove his conjecture, per the New York Times' Siobhan Roberts.

The team published a preprint paper detailing the findings on the site arXiv last week, and it has not been peer-reviewed yet. But experts say the work is expected to be supported with further investigation, per Science News.

"This appears to be a remarkable discovery," Joshua Socolar, a physicist at Duke University who did not contribute to the finding, tells the Times. "The most significant aspect for me is that the tiling does not clearly fall into any of the familiar classes of structures that we understand."

The "einstein" tile is made up of eight kites, or four-sided polygons with two pairs of adjacent, equal-length sides. Researchers call it "the hat" because of its resemblance to a fedora.

The shape is simpler than some experts expected it to be. Chaim Goodman-Strauss, a mathematician at the University of Arkansas and one of the authors of the paper, tells Science News that if he'd been asked to guess what the shape might look like before the finding, "I would've drawn some crazy, squiggly, nasty thing."

In the 1970s, mathematician Roger Penrose discovered that two shapes could form a non-repeating tiling pattern together, prompting hopes that a single shape may be found to do this one day. Researchers have been able to make other non-repeating patterns in the past, but the challenge has been finding a shape that can only make a non-repeating pattern, Goodman-Strauss tells the Times.

The shape of "the hat" can also be morphed to form additional tile shapes that make non-repeating patterns, as shown in the video above.

This new finding could lead to materials science investigations-for example, shapes that form non-repeating tilings could help design stronger materials, Hart tells New Scientist. The elusive shape might also spark creative inspiration for new decorative designs or art.

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At Long Last, Mathematicians Have Found a Shape With a Pattern That Never Repeats

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From bathroom floors to honeycombs or even groups of cells, tilings surround us. These patterns cover a space without overlapping or leaving any gaps. Like a rug filled with diamond shapes, where each section looks the same as the one next to it, every tiling ever recorded has eventually repeated itself-until now.

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Combine numbers to reach the target. New puzzles are available each day.

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10 Objects Mathematics: The Winton Gallery

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The question came from Claude Shannon, inventor of ' Information Theory ' in 1948. The theory uses mathematics to understand the rules governing the transmission of messages through communication systems, applicable to everything from computer code, speech and music, to the dancing of bees. Using maths and logic to understand the world around him, it wasn't long before Shannon began to wonder if a computer could beat a human at games, such as chess. In 1950 he wrote a paper asserting this possibility, but it wasn't until the 1970's that computers began to defeat humans at the game - generally poor players who made silly mistakes. But they could not defeat Grand-Masters. That did not happen until 1996 when DeepBlue beat Gary Kasparov. The following year the improved DeepBlue beat him 31/2-21/2.
So why did it take so long? Remember the question at the start?
There are between atoms in the observable universe. That's between ten quadrillion vigintillion and one-hundred thousand quadrillion vigintillion atoms. Which is a lot. But...amazingly, there are even more possible variations of chess games than there are atoms in the observable universe.

This is the Shannon Number and represents all of the possible move variations in the game of chess. It is estimated there are between positions (including illegal moves) in Chess. (If you rule out illegal moves that number drops dramatically to moves. Which is still a lot!).

"There are even more possible variations of chess games than there are atoms in the observable universe."

You might think, 'well a computer has conquered the most complicated game in the world there's nothing left for them to do?' and you'd be...wrong! There is a game with even more possible moves and variations and it is called Go. Thought to have originated in China over 4.000 years ago it did not become popular until it arrived in Japan around the year 500. It is played extensively in SE Asia: professionals start learning the game as very small children and spend all their lives perfecting their ability.

Go has more than moves...making it a googol times more complicated and varied than Chess and dwarfing the number of atoms in the Universe!
Do you think a computer or Artificial Intelligence could ever master a game this complicated in your lifetime?
Amazingly, it already has. Enter AlphaGo. In 2015 it played its first match against reigning three-time European Champion Mr. Fan Hui, and beat him 5-0.

In March 2016, the AI then competed against legendary Go player, eighteen-time world title winner Mr. Lee Sedol. It's said that Sedol is to Go what Federer is to tennis, yet, with 200 million people watching world wide, AlphaGo beat him 4-1 in a competition in Seoul, South Korea.

All Go players are ranked; an absolute beginner is ranked as Kyu 30. As they improve the move towards the rank Kyu 1. As they continue to improve they then join the Dan ranks, starting at level 1 and aim for (but rarely reach) level 9 Dan. There are currently just over 100 9 Dan players in the world. AlphaGo is one of them.
There's more...
The company that created AlphaGo - Deepmind - released a newer more powerful version, AlphaGo Zero.

According to Deepmind: "AlphaGo learnt Go by playing thousands of matches with amateur and professional players, AlphaGo Zero learnt by playing against itself, starting from completely random play...and then by playing against the strongest player in the world, AlphaGo.

This powerful technique is no longer constrained by the limits of human knowledge. Instead, the computer program accumulated thousands of years of human knowledge during a period of just a few days. Go Zero quickly surpassed the performance of all previous versions and also discovered new knowledge, developing unconventional strategies and creative new moves, including those which beat the World Go Champions Lee Sedol and Ke Jie..."

Now you may think:

"So what? A machine can play a game, big deal."

The big deal is that it has been able to make new discoveries, novel approaches. That AI can have 'creative moments' suggest that AI can be used to enhance human ingenuity rapidly.

When dealing with vast amounts of information, when attempting to understand lots of data (particularly mathematical) the human mind can become overwhelmed and tires quickly. An AI doesn't have those problems. AlphaGo Zero learnt thousands of years of human knowledge in just a few days. Applying that ability to other areas will allow patterns and discoveries that might otherwise be hidden or take a long time to discover by people alone.
How does an Artificial Intelligence learn?
That's a really good question and it is both simple and very clever. There are three levels of learning for an AI: Artificial Intelligence, machine learning, and deep learning.

Artificial Intelligence: is the lowest level of computer 'intelligence.' It mimics human learning by making decisions based on options and checks them with stored information: Is it round or curvy? Is it green or yellow? Is it a lime or a banana?

Machine Learning: comes from experience. Are all round green things limes? Can they be apples? Is it bigger than a certain size? Therefore, based on what it has 'learnt' from option choosing it can say what the object is.

Deep Learning: is a subset of Machine Learning, the software can train itself (using hidden layers called Neural networks) to understand its outputs. This approach uses huge amounts of data as it requires the machine to check with its database (experience) to find things it 'knows' already to allow it to identify objects.

"That AI can have 'creative moments' suggest that AI can be used to enhance human ingenuity rapidly."

What has this got to do with Astronomy?
Well I'm glad you asked, no really I am.

Even though there are more moves in Chess than atoms in the Universe, the Universe is still very, very big. It is estimated that there are 200-350 billion stars in our galaxy (the Milky Way). Our galaxy is a medium sized galaxy. It is believed there may be over a trillion galaxies in the visible universe and many more that we can't see.

Think of it this way; next time you go to the seaside, grab a handful of sand, or dig a hole in the sand. How many grains of sand do you think there are in your hand or in the pile you've just dug? Thousands? Millions maybe? Now look at the whole beach and try to guess how many grains there are.

It is thought that there are more stars in the universe than grains of sand on every beach on Earth. Most of those stars have at least one planet, often many more, orbiting. So there are even more planets than stars...

Astronomers and Astrophysicists deal with lots and lots of data and as technology improves the amount of data collected increases.

There are many new telescopes and observatories under development and soon to come on-line, as well as the space based James Webb Telescope and the Extremely Large Telescope in Chile there is also coming to Chile in 2021 the Large Synoptic Survey Telescope (LSST) also known as the Vera Rubin telescope.

When it begins operation it will take more than 800 panoramic images each night with a 3.2-billion-pixel camera, recording the entire visible sky twice each week.

Each night it will produce 20 TB of data. The images taken by the LSST Camera are so large that it would take 378 4K ultra-high-definition TV screens to display one of them in full size!

1TB is the same as 681 episodes of The Queen's Gambit (one can dream!).

SO much data, you say? You're right! It's far more than we humans can work on. That's just from one telescope...! There are a whole variety of programs, AI, Machine Learning and Deep Learning systems being used by Astronomers and researchers. Because it is more than a human can cope with, trainable neural networks are needed to help with classifying objects and suggesting to Astronomers those that might be interesting to look at more closely.

The European Southern Observatory has developed Morpheus: a deep-learning framework that incorporates a variety of artificial intelligence technologies developed for applications such as image and speech recognition. To help astronomers, Morpheus will work pixel by pixel through the images looking for galaxies! An Older Morpheus result from 2016, working with Hubble, revealed that here were 10 times more galaxies than previously thought.

"An Older Morpheus result from 2016, working with Hubble, revealed that here were 10 times more galaxies than previously thought."

Researchers at Lancaster University have developed at system called Deep-CEE (Deep Learning for Galaxy Cluster Extraction and Evaluation), a novel deep learning technique to speed up the process of finding galaxy clusters.

First discovered in 1950 by George Abell, galaxy clusters are rare but massive objects. Abell spent years scanning 2000 photographic plates with his eye and a magnifying glass and found 2,712 clusters. Galaxy clusters are important as they will help us understand how dark matter and dark energy have shaped our universe.

Deep-CEE builds on Abell's approach replacing the astronomer with an AI model trained to "look" at colour images and identify galaxy clusters. It is a state-of-the-art model based on neural networks, which are designed to mimic the way a human brain learns to recognise objects by activating specific neurons when visualizing distinctive patterns and colours. The AI was trained by repeatedly showing it examples of known, labelled, objects in images until the algorithm learnt to recognise objects on its own.

Deep-CEE will also be used on the Rubin telescope.

Not yet finished (but with Phase 1 already running) is the Square Kilometre Array (SKA); a series of radio telescopes that span continents and will be the largest ever radio telescope. It's headquarters are in Jodrell Bank in Cheshire.

The majority of the telescopes will be in South Africa and Australia. It will need 2 super-computers to handle all the data. In South Africa there will be 197 radio dishes and in Australia over 131,000 antennae!

Each year SKA will amass 600 PetaBytes of data (or 1.6 PB per day or ~630 Netflix videos a day). To store this data on an average 500 GB laptop, you would need more than a million of them every year. 500GB is the equivalent of 500 hundred lorries full of paper.
But why does the SKA need such immense computing power?
Scientific image and signal processing for radio astronomy consists of several fundamental steps, all of which must be completed as quickly as possible across thousands of telescopes connected by thousands of miles of fibre optic cable. The computers must be able to make decisions on objects of interest, and remove data which is of no scientific benefit, such as radio interference from things like mobile phones.

What about all the rest?

Then of course there are the telescopes, observatories and satellites that are already working: perhaps one of the most famous is the Hubble Space Telescope.

Hubble transmits about 120 gigabytes of science data every week. That would be roughly 1,097 metres (3,600 feet) of books on a shelf. Hubble has been operational for 30 years and has made over 1.5 million observations.

It's not just about the data, to get that you need to schedule observations. This can be incredibly complicated - timing, location of object, position of spacecraft, rising and setting times and many other variables have to be looked at. To organise observations and timings Hubble uses SPIKE which uses a very fast neural network-inspired, scheduling algorithm to achieve performance humans can only dream of.

We may not have smart cars or personal robots but advances in Artificial Intelligence are already providing profound benefits and discoveries for us all. AI is not our master, it can only learn based on how we program it and what we determine as important...at least for now!

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Which is greater? The number of atoms in the universe or the number of chess moves?

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The question came from Claude Shannon, inventor of 'Information Theory' in 1948. The theory uses mathematics to understand the rules governing the transmission of messages through communication systems, applicable to everything from computer code, speech and music, to the dancing of bees.

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How to Solve Math Questions! | Math Hacks and Tricks by Blossom

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The mathematical secrets of Pascal’s triangle - Wajdi Mohamed Ratemi

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Pascal’s triangle, which at first may just look like a neatly arranged stack of numbers, is actually a mathematical treasure trove. But what about it has so intrigued mathematicians the world over? Wajdi Mohamed Ratemi shows how Pascal’s triangle is full of patterns and secrets.

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Fibonacci sequence is found by adding the previous two numbers of the sequence together. Have you spotted this in nature?

Fibonacci (real name Leonardo Bonacci) was a mathematician who developed the Fibonacci Sequence. The sequence is found by adding the previous two numbers of the sequence together. It looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... And on it goes. This pattern can also be seen as:

The Fibonacci Sequence is found all throughout nature, too. It is a naturally occurring pattern. Here are some examples of Fibonacci in nature...

Tree Branches

Although we all usually see trees everywhere in our day to day life, how often have you looked for the patterns in them? In trees, the Fibonacci begins in the growth of the trunk and then spirals outward as the tree gets larger and taller.

We also see the golden ratio in their branches as they start off with one trunk which splits into 2, then one of the new branches stems into 2, and this pattern continues.

Storms

Your eye of the storm is like the 0 or 1 in the Fibonacci sequence, as you go on in the counter-clockwise spiral you find it increasing at a consistent pattern. This pattern is much like the Golden Ratio. But is a hurricane actually a Fibonacci spiral?? >>

Seashells

When cut open, nautilus shells form a logarithmic spiral, composed of chambered sections called camerae. Each new chamber is equal to the size of the two camerae before it, which creates the logarithmic spiral. This proportional growth occurs because the nautilus grows at a constant rate throughout its life until reaching its full size.

Flower Petals

The petals of a flower grow in a manner consistent with the Fibonacci. Of the most visible Fibonacci sequence in plants, lilies, which have three petals, and buttercups, with their five petals, are some of the most easily recognised.

Galaxies

The golden spiral can be found in the shape of the "arms" of galaxies if you look closely. It can't be told if galaxies follow a perfect spiral, because we can't measure a galaxy accurately, but on paper, we can measure it and see the size. Read more on Fibonacci in galaxies here >>

Flower Heads

Most of the time, seeds come from the centre of the flower head and migrate out. A perfect example of this is sunflowers with their spiralling patterns. At points, their seed heads get so packed that their number can get extremely high, sometimes as much as 144 and more. When analysing these spirals, the number is almost always Fibonacci.

YOU!

You are an example of the beauty of the Fibonacci Sequence. The human body has various representations of the Fibonacci Sequence proportions, from your face to your ear to your hands. You have now been proven to be mathematically gorgeous.

There are so many more examples of Fibonacci in nature we couldn't include in this list. Read more about Fibonacci on Odyssey >>

Interested in STEM? Want to find like-minded peers? Join us in the Stemettes Society.

Are you a young woman or a non-binary young person looking for a mentor for a headstart in your STEM career? Check out our STS programme.
Image credits: Giulia May | Unsplash

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How Many Times Have You Spotted Fibonacci In Nature? Here Are 7 Examples For You... - The Stemettes Zine

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Fibonacci sequence is found by adding the previous two numbers of the sequence together. Have you spotted this in nature? Fibonacci (real name Leonardo Bonacci) was a mathematician who developed the Fibonacci Sequence. The sequence is found by adding the previous two numbers of the sequence together.

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An explanation of the strangest problem in maths.

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Arthur Benjamin: The magic of Fibonacci numbers

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Math is logical, functional and just ... awesome. Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series. (And reminds you that mathematics can be inspiring, too!)

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Fill in the different colors to match the different ratios!

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Scale of the Universe - A scale bar for the universe

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Scale of the Universe - A scale bar for the universe

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Scale of the Universe: Discover the vast ranges of our visible and invisible world.

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Scale of Universe is an interactive experience to inspire people to learn about the vast ranges of the visible and invisible world.

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Our flagship game, nerdle, was inspired by Wordle and a passion for math. It challenges you to find the hidden calculation in just six guesses. In the wider nerdleverse, you'll roam an entire galaxy of puzzles, including number-based crosswords, anagrams, and so much more. Whether you're a math enthusiast or just looking for a daily brain workout, the nerdleverse has something special for you. Join us and let your mind explore the wonders of the nerdleverse!

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Nerdle - guess the solution in 6 tries - try classic, mini, speed and pro

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Use mathematical logic to tackle some spy puzzles!

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Use mathematical logic to tackle some spy puzzles!

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Solving the seemingly impossible is part and parcel of GCHQ's work to keep the country safe. Many of their workforce like to keep their minds sharp by both creating and solving puzzles for fun!

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Solving the seemingly impossible is part and parcel of GCHQ's work to keep the country safe. Many of our workforce like to keep their minds sharp by both creating and solving puzzles for fun!