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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Miura Fold

Imagine that we want to launch a satellite into space, which, once in orbit, will be powered by rigid solar panels that fan outward. But, to launch the satellite the panels have to be folded and compact. How would we design them?

In 1985, Koryo Miura, a Japanese astrophysicist, proposed a method of folding a flat surface into a smaller area: the Miura fold. This folds paper or other materials in such a way that allows each section to remain flat, which is a necessary condition for stiff materials like solar panels. These folds are considered to be shape-memory origami as after unfolding, the sheet can easily be re-folded and returned to its more compact shape, and hence the fold can be ‘remembered‘.

 

File:Miura-ori.gif

Miura Fold | Source: Wikipedia

The Miura fold was used in Japan’s Space Flyer Unit, launched in 1995, and has “influenced the development of other folds that allow materials to be packed into a compact shape and then unfold in one continuous motion.”

The crease patterns of the Miura fold form a tessellation of the surface by parallelograms.

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Source: sciencefriday

In one direction the creases lie along straight lines, where each parallelogram forms the mirror reflection of its neighbour along the crease. However, in the other direction, the creases zigzag and each parallelogram is the translation of its neighbour along the crease.

The Miura fold is a form of rigid origami: “the fold can be carried out by a continuous motion in which, at each step, each parallelogram is completely flat.

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Kakeya Needle Problem

The Kakeya needle problem asks whether there is a minimum area of a region in the plane in which a line segment of width 1 can be freely rotated through 360°, where translation of the segment is allowed.

This question was first posed for convex regions in 1917 by mathematician Sōichi Kakeya. It was shown by Gyula Pál that the minimum area for convex regions is achieved by an equilateral triangle of height 1 and area 1/√3.

Kakeya suggested that the minimum area, without the convexity restriction, would be a three-pointed deltoid shape. However, this is false.

Needle rotating inside a deltoid | Source: Wikipedia

Besicovitch Sets

Besicovitch was able to show that there is no lower bound >0 for the area of a region in which a needle of unit length can be turned around. The proof of this relies on the construction of what is now known as a Besicovitch set, which is a set of measure zero in the plane which contains a unit line segment in every direction.

One can construct a set in which a unit line segment can be rotated continuously through 180 degrees from a Besicovitch set consisting of Perron trees.

Kakeya Needle Set constructed from Perron trees | Source: Wikipedia

However, although there are Kakeya needle sets of arbitrarily small positive measure and Besicovich sets of measure 0, there are no Kakeya needle sets of measure 0.

Video: Numberphile

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MATHS BITE: Dini’s Surface

Dini’s surface, named after Ulisse Dini, is a surface with constant negative curvature that can be created by twisting a pseudosphere (see picture below).

Pseudosphere

Dini’s surface is given by the following parametric equations:

{\displaystyle {\begin{aligned}x&=a\cos u\sin v\\y&=a\sin u\sin v\\z&=a\left(\cos v+\ln \tan {\frac {v}{2}}\right)+bu\end{aligned}}}

 

 

Dini’s Surface

Dini’s surface is pictured in the upper right-hand corner of a book by Alfred Gray (1997), as well as on the cover of volume 2, number 3 of La Gaceta de la Real Sociedad Matemática Española (1999).

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NEWS: Abel Prize 2017

The Abel Prize 2017 has been awarded to Yves Meyer of the École normale supérieure Paris-Saclay in France due to his “pivotal role in the development of the mathematical theory of wavelets”, which has applications in data compression, medical imagery and the detection of gravitational waves.

Yves Meyer, en 2010, recevant le prix Gauss.

Meyer, aged 77, will receive 6 million Norwegian krone (around £600,000) for the prize, which aims to recognise outstanding contributions to mathematics. It is often called the ‘Nobel Prize’ of mathematics.

The Abel Prize was previously won by Andrew Wiles in 2016, who solved Fermat’s Last Theorem.

Biography

Yves Meyer, born on the 19th July 1939, grew up in Tunis in the North of Africa. After graduating from École normale supérieure de la rue d’Ulm in Paris and completing a PhD in 1966 at the University of Strasbourg, he became a professor of mathematics at the Université Paris-Sud, then the École Polytechnique and then Université Paris-Dauphine. He then moved to École normale supérieure Paris-Saclay in 1995, until formally retiring in 2008, although he still remains an associate member of the research centre.

To read a full biography of Meyer, click here.

Video of the Ceremony

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Nine-Point Circle

The nine-point circle is a circle that can be that can be constructed for any given triangle. It is named the nine-point circle as it passes through nine points defined from the triangle:

  • The midpoint of each side of the triangle (Ma, Mb, Mc);
  • The foot of each altitude (Ha, Hb, Hc);
  • The midpoint of the line segment from each vertex of the triangle to point where the three altitudes meet, i.e. the orthocentre H, (Ea, Eb, Ec).
Nine-PointCircle

Nine Point Circle | Source: Wolfram Mathworld

Note that for an acute triangle, six of the points – the midpoints and altitude feet – lie on the triangle itself. On the contrary, for an obtuse triangle two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point triangle.

The nine-point circle is the complement to the circumcircle, which is the unique circle that passes through each of the triangle’s three vertices.

Circumcircle

Circumcircle | Source: Wolfram Mathworld

Although credited for its discovery, Karl Wilhelm Feuerbach did not entirely discover the nine-point circle, but rather the six point circle, as he only recognised that the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle lay on the circle. It was mathematician Olry Terquem who was the first to recognise the added significance of the three midpoints between the triangle’s vertices and the orthocenter.

Three Properties of the Nine-Point Triangle

  • The radius of a triangle’s circumcircle is twice the radius of the same triangle’s nine-point circle.
9pcircle03.svg

Source: Wikipedia

  • A nine-point circle bisects a line segment going from the corresponding triangle’s orthocenter to any point on its circumcircle.
9pcircle 04.png

Source: Wikipedia

  • All triangles inscribed in a given circle and having the same orthocenter have the same nine-point circle.

Want to know more?

Click here to find out to construct a nine-point circle and here to read a quick proof of its existence!

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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