Why doesn’t any animal have three legs?

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If ‘Why?’ is the first question in science, ‘Why not?’ must be a close second. Sometimes it’s worth thinking about why something does not exist. Such as a truly three-legged animal. At least one researcher has been pondering the non-existence of tripeds.

“Almost all animals are bilateral,” he said. The code for having two sides to everything seems to have got embedded in our DNA very early in the evolution of life — perhaps before appendages like legs, fins or flippers even evolved. Once that trait for bilateral symmetry was baked in, it was hard to change.

With our built-in bias to two-handedness, it can be hard to figure out how a truly three-legged animal would work — although that has not stopped science fiction writers from imagining them. Perhaps trilateral life has evolved on Enceladus or Alpha Centauri (or Mars!) and has as much difficulty thinking about two-limbed locomotion as we do thinking about three.

This kind of thought experiment is useful for developing our ideas about evolution, Thomson said. (1)

How fascinating.

Everything started with Nothing.

Then One came into existence.

We are still in the phase of Two…

And there is no way to get any further.

For going further means that we get to three.

And from there infinity is one step away.

Leading to nothing more than zero once again…

But there is no infinity.

There is no two.

Not even One.

For only everything exists.

Infinity!

In the palm of a small kid…

Chaos. Numbers. Simulations.

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Digital computers use numbers based on flawed representations of real numbers, which may lead to inaccuracies when simulating the motion of molecules, weather systems and fluids, find scientists.

The study, published today in Advanced Theory and Simulations, shows that digital computers cannot reliably reproduce the behaviour of ‘chaotic systems’ which are widespread. This fundamental limitation could have implications for high performance computation (HPC) and for applications of machine learning to HPC.

Professor Peter Coveney, Director of the UCL Centre for Computational Science and study co-author, said: “Our work shows that the behaviour of the chaotic dynamical systems is richer than any digital computer can capture. Chaos is more commonplace than many people may realise and even for very simple chaotic systems, numbers used by digital computers can lead to errors that are not obvious but can have a big impact. Ultimately, computers can’t simulate everything.”

The team investigated the impact of using floating-point arithmetic — a method standardised by the IEEE and used since the 1950s to approximate real numbers on digital computers.

Digital computers use only rational numbers, ones that can be expressed as fractions. Moreover the denominator of these fractions must be a power of two, such as 2, 4, 8, 16, etc. There are infinitely more real numbers that cannot be expressed this way. (https://www.sciencedaily.com/releases/2019/09/190923213314.htm)

An irrational universe.

Full of irrational people.

Trying to analyze it rationally.

Under the illusion that number we have invented can draw a sketch of the cosmos. And yet, nothing we have invented is anywhere to be seen but on a piece of paper. Can you limit the birth of a star on a piece of paper? Can you contain the death of the universe on an equation?

We believe we can.

And sadly, we do.

And at the moment we do, the universe indeed dies…

And a small voice will whisper in our ear…

Congratulations. You have now understood it all.

How irrationally rational everything is!

And inside the darkest night you will dance.

Laughter.

And for a brief moment the forest will look at you.

Crying.

And for a brief moment the forest will see nothing…

But an empty broken CD. Full of data. Full of life…

Old mathematics… Broken cosmos… Blurry image…

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By combining cutting-edge machine learning with 19th-century mathematics, a Worcester Polytechnic Institute (WPI) mathematician worked to make NASA spacecraft lighter and more damage tolerant by developing methods to detect imperfections in carbon nanomaterials used to make composite rocket fuel tanks and other spacecraft structures.

Using machine learning, neural networks, and an old mathematical equation, Randy Paffenroth has developed an algorithm that will significantly enhance the resolution of density scanning systems that are used to detect flaws in carbon nanotube materials.

The algorithm was “trained” on thousands of sets of nanomaterial images and to make it more effective at making a high-resolution image out of a low-resolution image, he combined it with the Fourier Transform, a mathematical tool devised in the early 1800s that can be used to break down an image into its individual components.

“The Fourier Transform makes creating a high-resolution image a much easier problem by breaking down the data that makes up the image. Think of the Fourier Transform as a set of eyeglasses for the neural network. It makes blurry things clear to the algorithm. We’re taking computer vision and virtually putting glasses on it”, said Paffenroth. (1)

We like breaking the world into pieces.

We can see better that way.

But even the sharpest image of a tree.

Conveys nothing about the forest…

A forest that is there because of the trees.

Trees we know are there.

We remember those trees.

We once saw those trees.

Casting their shadows during the evening hours.

At a time when we used to stand within a forest.

But never really saw one…

Cause in the midst of the evening.

There was nothing else casting a shadow.

Nothing but our self!

1+1 = ?

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Researchers from UNIGE and the University of Bourgogne Franche-Comté tested the degree to which our worldly knowledge and perspective interferes with mathematical reasoning by presenting twelve problems to two distinct groups. The first group consisted of adults who had taken a standard university course, while the second was composed of high-level mathematicians.

When faced with numbers, we tend to represent them mentally either as sets or as values on axes. “We devised six 5th grade subtraction problems (i.e. for pupils aged 10-11) that could be represented by sets, and six others that could be represented by axes,” begins Emmanuel Sander, an FPSE professor. “But all of them had exactly the same mathematical structure, the same numerical values and the same solution. Only the context was different.” Half of the problems involved elements that can be grouped together as sets (e.g. calculating the number of animals in a pack). For example: “Sarah has 14 animals: cats and dogs. Mehdi has two cats fewer than Sarah, and as many dogs. How many animals does Mehdi have?” The second type of problems involved elements that can be represented along a horizontal or vertical axis (e.g. calculate how tall a Smurf is). For example: “When Lazy Smurf climbs onto a table, he attains 14 cm. Grumpy Smurf is 2 cm shorter than Lazy Smurf, and he climbs onto the same table. What height does Grumpy Smurf attain.”

These problems can all be solved via a simple subtraction. “This is instinctive for the problems represented on an axis (14 – 2 = 12, in the case of the Smurfs) but we need to change perspective for the problems describing sets. For instance, in the problem with animals, we look to calculate the number of dogs that Sarah has, which is impossible, whereas the calculation 14 – 2 = 12 provides the solution directly,” explains Jean-Pierre Thibaut, a researcher at the University of Bourgogne Franche-Comté. The scientists relied on the fact that the answer would be more difficult to find for the animal problems than the Smurf problems, despite their shared mathematical structure.

The results were astonishing. In the non-expert adult group, 82% answered correctly for the axis problems, compared to only 47% for the problems involving sets. Regarding the expert mathematicians, 95% answered correctly for the axis problems, a rate that dropped to only 76% for the sets problems! “One out of four times, the experts thought there was no solution to the problem even though it was of primary school level! And we even showed that the participants who found the solution to the set problems were still influenced by their set-based outlook, because they were slower to solve these problems than the axis problems,” continues the Geneva-based researcher.

The results highlight the critical impact our knowledge about the world has on our ability to use mathematical reasoning. They show that it is not easy to change perspective when solving a problem. (1)

Think too much.

And at the end you will stop thinking.

Learn things.

And at the end you will forget what you knew.

Once upon a time you knew how to divide numbers.

But then you tried to learn how to multiply.

Once upon a time you knew how to subtract numbers.

But then you tried to learn how to add.

Once upon a time you knew how to count.

But then you decided to learn how numbers are.

Once upon a time you knew how to draw a circle.

But then you tried to learn how to make a rectangle.

New knowledge constantly added. Only because future knowledge comes. Our constant thirst for new discoveries drives our way through the realms of science. And as we move on, the new knowledge only adds to our ignorance. And the knowledge that has yet to come justifies the knowledge we currently get and clarify.

We have created math to describe the cosmos. But from the moment we started counting the world started laughing. For what is to be known will never be. The veil of mystery for the true meaning of existence is not lifted up with what we do. With every question and every equation we do not tear up that veil in order to see behind it. We are just weaving on it. And it will soon be complete opaque.

In a vicious circle of endless thinking, we can only think of moving forward. But only because we have already been at the end.

Yes, we did first learn to add and then to subtract.

But only because subtraction is addition.

In the same way addition is subtraction.

There is nothing new to learn besides the things you have already learned.

And at the end, when all mathematical operations, have been completed.

You will see zero on both sides of the equation.

And you will remember smiling…

That this is where you started crying…

Changing geometry. Blurry lines…

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Atomic interactions in everyday solids and liquids are so complex that some of these materials’ properties continue to elude physicists’ understanding. Solving the problems mathematically is beyond the capabilities of modern computers, so scientists at Princeton University have turned to an unusual branch of geometry instead.

Researchers led by Andrew Houck, a professor of electrical engineering, have built an electronic array on a microchip that simulates particle interactions in a hyperbolic plane, a geometric surface in which space curves away from itself at every point. A hyperbolic plane is difficult to envision — the artist M.C. Escher used hyperbolic geometry in many of his mind-bending pieces — but is perfect for answering questions about particle interactions and other challenging mathematical questions. (1)

Draw a line on the paper.

Look at the circle on the sand.

A teardrop falling on water.

The moon circling the Earth.

A circle turning into a square.

Sun turning into darkness.

The ink is blurring now.

The line is fading.

And with strange aeons…

Even the paper will reduce into dust.

Your geometry will be lost. Along with everything reminding it. You will be alone at the end. And your tears will fall in the water. And they will create circles again. Don’t cry. Just take the pen. Don’t wander whether you can draw one on paper. You know you can…

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