Pictures of Math VII

This post was inspired by an article I read on sciencealert.com.

Simon Beck’s Snow Art

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Simon Beck is the world’s first snow artist. Each pattern takes him 11 hours, and he uses nothing more than a compass and his snowshoes. He chooses to draw maths due to the simplicity of the patterns.

Fabergé Fractals

UK physicist Tom Beddard decided to create digital renderings of 3D Fabergé eggs covered in fractal patterns.

“The formulae effectively fold, scale, rotate or flip space. They are truly fractal in the fact that more and more detail can be revealed the closer to the surface you travel.” – Beddard

For more read here.

3D Models by Henry Segerman

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Henry Segerman is an Australian mathematician who creates 3D printed models that express mathematical formulae and concepts.  This allows his students to better understand them.

“The language of mathematics is often less accessible than the language of art, but I can try to translate from one to the other, producing a picture or sculpture that expresses a mathematical idea.” 

Hevea Project

A French team of mathematicians called the Hevea Project, have created digital constructions of isometric embeddings.

“Take a sphere – say the surface of a tennis ball – and imagine shrinking it down to have a nanometre radius,” writes Daniel Matthews about isometric embedding. “Nash and Kuiper show that by ‘ruffling’ the surface sufficiently (but always smoothly; no creasing or folding or ripping or tearing allowed!) you can have an isometric copy of your original tennis ball, all contained within this nanometre radius.”

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VIDEO: Spiral Sculptures

John Edmark is an artist and professor at Stanford University who has used the Golden Angle to sculpt spirals. The Golden Angle is derived from the Golden Ratio: it is the smaller of the two angles created by dividing the circumference of a circle according to the golden ratio and comes out to be around 137.5°.

For more click here.

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Ulam Spiral

The Ulam Spiral, discovered in 1963 by Stanislaw Ulam, is a graphical depiction of the set of prime numbers.

If you were to arrange the positive numbers in a spiral, starting with one at the centre, then circle all of the prime numbers, what would you get? As prime numbers don’t have a predictive structure, you would expect to get little or even nothing out of arranging the primes this way. But, Ulam discovered something incredible:

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Ulam Spiral

To his surprise, the circled numbers tended to line up along diagonal lines. In the 200×200 Ulam spiral shown above, diagonal lines are clearly visible, confirming the pattern. Although less prominent, horizontal and vertical lines can also be seen.

Even more amazing, this pattern still appears even if we don’t start with 1 at the centre!

There are many patterns on this plot. One of the simplest ones is that there are many integer constants b and c such that the function:

f(n) = 4 n^2 + b n + c
generates, a number of primes that is large by comparison with the proportion of primes among numbers of similar magnitude, as n counts up {1, 2, 3, …}.

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Pictures of Maths VI

Images by by Samuel Monnier visually exploring the Lattès map:

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The golden lines highlight the complexity of turbulent convective flow. They have been created using a numerical simulation of turbulent Rayleigh-Benard convection, which is when the fluid is trapped between two plates and is heated from above and cooled from below. This visualisation shows skin friction on the bottom (heated) plate in a flow of turbulently convecting liquid mercury. The bright lines are areas with large velocity changes at the wall, showing high shear stress and vigorous convective flow.

Geometric patterns on the ceilings of European cathedrals.

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By John M. Sullivan

The Weaire-Phelan foam is a counterexample to Kelvin’s Conjecture about the best partition of space into equal-volume cells.

The interactions between sub-atomic particles at the particle accelerator at CERN in Geneva.

To see the previous posts in this series, click here! M x

Mandelbrot and Julia Sets

In today’s post I wanted to quickly highlight a cool relationship between Mandelbrot and Julia sets.

Consider the function, which depends of complex parameter z:

f(z) = x2 z 

Fixing this z, f(z) defines a map from the complex plane to itself. We can start from any value of x and apply this function over and over, which would give us a sequence of numbers. This sequence can either go off to infinity, or not. The boundary of the set of values of x where it doesn’t is the Julia set for this particular z, which we fixed initially.

Conversely, starting with x = 0, we can draw the set of numbers for which the resulting sequence does not go off to infinity. This is called the Mandelbrot set. (Note the subtle difference between the two).

Okay, so the cool relationship is that, near the number z, the Mandelbrot looks like the Julia set for the number z, or as Wikipedia describes:

“There is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set.”

To illustrate this, consider the following Julia set:

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Zooming into the Mandelbrot set at the same value of z gives us this image:

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They are extremely similar! So, essentially, the Mandelbrot sets looks like a lot of Julia sets! (Click here to explore this in more detail).

This amazing result is used in lots of results on the Mandelbrot set, for example, it was exploited by Shishikura to prove that “for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane“.

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Love & Math

Edward Frenkel is a Russian mathematician and professor at University of California, Berkley. However, he is perhaps most well known for his best-selling book ‘Love and Math: The Heart of Hidden Reality’, which was published in October 2013.

Frenkel believes that the way mathematics is taught in schools today is “about as exciting as watching paint dry“, and as a result most people have a very bad experience with it; as a mathematics student, when people ask me what I study I am often met with the response of “oh god!”, “why?” or simply just a look of bewilderment.

Love and Math attempts to show a side of mathematics that we’ve never seen. It tells two intertwined stories: of the wonders of mathematics and of one man’s journey in learning and living it. In this way, this book shows the reader how to access a new way of thinking and invites them to discover “the hidden magic universe of mathematics“.

Frenkel insists that there is no need to ‘suffer’ through years of mathematical study to grasp the key ideas, but rather we can learn “a few chords“, he says. So what are these ‘chords’? For example, instead of learning something “dry” like Euclidean geometry, we should learn the concept of symmetry. Whilst it may seem like a simple idea, the mathematics involving symmetry can quickly grow more elaborate, and in fact it played a fundamental role in the discovery of quarks, which are the elementary particles that protons and neutrons are composed of.

If you didn’t enjoy mathematics in secondary school, the following episode of the Inquiring Minds podcast, featuring Edward Frenkel, may catch your attention.

For more on Edward Frenkel click here or here.

My experience with mathematics in secondary school was a rather negative one in the sense that I never had an amazing teacher who transmitted a real passion for the subject to their students. Instead, my lessons were often filled with a 10 minute (very dry) powerpoint at the start to explain the content (but just what we needed to know for the exam), and then 50 minutes of answering very routine questions. We did not get exposed to the level of problem solving that I am now being shown in university, nor did we really learn the fundamentals of mathematics, for example what does it mean to differentiate or integrate? I was never even taught what a limit was, I had to teach that to myself! Due to this, I feel like a lot of students where pushed away from perhaps pursuing mathematics, which is a real shame. I was lucky to always have this inner passion for mathematics, which was the driving force throughout my GCSEs and A-Levels.

What are your thoughts on mathematics in high school/ secondary school? M x

Sublime Symmetry

The De Morgan Foundation organised a one day synopsium called ‘Sublime Symmetry’ on the 13th January, which explored the mathematics behind William De Morgan’s ceramic designs.

William De Morgan was a ceramic designer in the late Victorian period.

“His conjuring of fantastical beasts to wrap themselves around the contours of ceramic hollowware and his manipulation of fanciful flora and fauna to meander across tile panels fascinated his contemporaries and still captivates today.

The ‘Sublime Symmetry’ exhibition highlights the influence of geometry in William De Morgan’s work, and particularly the use of symmetry to create his designs. This application of geometry naturally produces beautiful and visually striking images. Below are some images of his work:

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Source: auckboro.wordpress.com

I find it fascinating how mathematics is so naturally interlaced in art and beauty, and so I really wanted to share this with you. Hopefully you found it interesting as well! The ‘Sublime Symmetry’ exhibition will be at the New Walk Gallery in Leicester until the 4th March, after which it will be displayed at the William Morris Gallery in Walthamstow on the 12th March until the 3rd September. [Source]

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Pictures of Maths V

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This is a tessellation of Penrose tiles. Penrose tiles can be arranged, as has been done in this image, such that the tiling never repeats, no matter how many tiles are used.The two different tile shapes are a wider rhombus and a thinner rhombus. Additionally, each tile has been filled with four pursuit curves: the dark curves from each corner to a point neat the centre of the tile.

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This image was created by iterating systems of Möbius transformations. A Möbius transformation in the complex plane is a rational function of the form

f(z)={\frac  {az+b}{cz+d}}

where z is a complex variable, and a, b, c, d are complex numbers satisfying adbc ≠ 0.

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This double knit scarf brings together two mathematical ideas: a recursive construction of a fractal – the dragon fractal, and the recursive construction of an integer sequence – the Fibonacci sequence. The main panels of the scarf are based on a pattern which arises from the 11th iteration of the dragon fractal, whilst the striping between the main panels illustrates the Fibonnaci sequence (colour changes after 1 row, after 1 row, after 2 rows, after 3 rows, after 5 rows etc).

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This image shows 8,000 ellipses. For each k=1,2,3,…,8000 the foci of the kth ellipse are:

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The eccentricity of the kth ellipse is D(k), where

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This is part of the generating tile of a planar repeating pattern that have no reflection symmetries but do have many glide reflection symmetries as well as translational symmetries and two-fold centres of rotation. “The absence of reflectional symmetries often leads to very fluid and dynamic patterns.”

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Fibonacci Sequence with Differential Equations

Today I thought I’d talk about something I recently did in one of my lectures.

As most of you probably already know, the Fibonacci sequence is defined in the following way:

{\displaystyle F_{n}=F_{n-1}+F_{n-2},}

with initial values  {\displaystyle F_{1}=1,\;F_{2}=1}.

This sequence comes up everywhere, for example in biological systems describing the number of petals and the shape of broccoli.

Now let us solve the equation

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As this this is a difference equation, we can solve it using techniques for differential equations. First, we make the ansatz (this is just fancy German word for ‘educated guess’) that

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This will give us the simple equation: Screen Shot 2016-11-10 at 8.28.26 AM.png

This, as you can probably recognise, means that k is the golden ratio and its inverse!

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“For reasons that have still not been thoroughly resolved by neuroscientists, we are conditioned to find aesthetic appeal in structures with aspect ratios close to the golden ratio, a fact known empirically to artists and engineers from ancient times. One everyday example is the proportion of pieces of paper.” – Dr. C.P Caulfield 

Therefore, we know that the solution to the difference equation is

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Using the initial conditions, we can find A and B:

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Note that this is an expression for a sequence of integers in terms of differences in the powers of the golden mean, which is certainly an irrational number (in fact it can be argued to be the most irrational number of all as it can be expressed in terms of the convergence properties of its continued fraction representation)!

Also, as φ1 > 1 and rearranging the above expression, we can see that:

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So the ratio of consecutive numbers in the Fibonacci Sequence tends to the Golden Mean as n gets very large. Isn’t that amazing!

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Pictures of Maths IV

Greenfield
Untitled by Gary Greenfield

This image is produced by visualising the foraging behaviour of the ant P. barbatus. 1000 ants stream out of the nest along 6 main trails and break off to individually search for seeds. The image only highlights 500 of their steps. The colour graduations distinguish the “search phase from the return phase and reinforce the dynamic aspects of the process”.

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“Bunny” by Craig S. Kaplan and Henry Segerman

This sculpture is self-referential; it is a sculpture of a bunny whose surface is tiled by 72 copies of the word ‘bunny’. It is an example of an autologlyph, which is “a word written or represented in a way which is described by the word itself”. This type of autologlyph combines “Escher-like tessellation with typographical ideas related to ambigrams“.

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Intrinsic Transformation I by Conan Chadbourne

This work shows the structure of the icosahedral group. This is the smallest non-abelian simple group and is the set of “orientation-preserving symmetries of the regular icosahedron and dodecahedron”. The groups elements are shown as yellow disks arranged at the vertices of a truncated icosahedron, whilst the group’s generators – orders 2 and 5, which are coloured red and blue respectively – are depicted by the regions between the disks.

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Visualisation of the musical piece “Five Armies” by Kevin MacLeod

Musical Flocks is a project which produces animations by simulating the behaviour of agents that react to the sound of music. Slow musics results in the slow and gentle movement of the flock, whilst fast tempo music results in fast movement and abrupt changes. Sounds which have a high volumes and a “rich” frequency spectrum affect the majority of the boids, whereas low volume and less active frequencies gives “more subtle visual variations and slower graphic evolution”.

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Touch-graph by Annie Verhoeven

Verhoeven used ‘thread art’ to create the sculpture shown above. It visualises a computer program that is used to show connections between people; the coloured lines symbolise the different types of connections between people. “All these different connections are used in our life on every level, be it scientific, religious, political or otherwise, and can help us determine where we stand in the world.”

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Stars of the Mind’s Sky by Paul Salomon

In this image, 300 starts are positioned along concentric circles. The number of points on the star increases as the radius from the centre increases, and the stars with the same number of points are placed evenly along their circle according to the density (“jump number”) used to generate them. Mathematically, this can be seen as representing subgroups and cosets generated by elements of a cyclic group. The different colours highlight the number of cosets, with the red stars representing the generating element. Thus, “we may observe congruent stars with increasingly many cosets, shifting their way to blue along central rays through any red star.”

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The Yang-Baxter scarf by Robin Endelman

The scarf above depicts the Yang-Baxter equation in statistical mechanics. A variation of this equation is used in braid theory and in the 3rd Reidemeister move in knot theory. In this scarf, the numbers 1, 2 and 3 are assigned to the colours blue, green and gold respectively. The Yang-Baxter equation states that R12 R13 R23 = R23 R13 R12, where Rij denotes the point where strand i crosses strand j; the two sides of the equation are highlighted by the two ends of the scarf, and the middle section demonstrates the equality (=).

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Hyperbolic Fractal Tiling 1 by Robert Fathauer

This image shows four different views of the same 3D object – a fractal tiling where every tile has a similar dart shape. This demonstrates how a complex organic structure can be created from the repeated application of a simple set of rules to a simple starting structure.

Image credits to Discover magazine.

Read my previous posts in this series here. M x