geometry

Proof Without Words #3

Proof of the identity

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The figure for general n is similar, with n right pyramids, one with an (n-1)-cube of side length xk as its base and height xk for each k=1,….,n.


The derivative of sin is cosine.

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From ‘Proof without words‘ by Roger Nelsen

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By Sidney H. Kung

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By Shirley Wakin


Previous ‘Proof Without Words‘: Part 1 | Part 2

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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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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.
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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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Intrinsic Geometry

Today I wanted to discuss the geometry of curves and surfaces.

Curves, Curvature and Normals

First let us consider a curve r(s) which is parameterised by s, the arc length.

Now, t(s) = Screen Shot 2017-02-14 at 8.05.47 AM.png is a unit tangent vector and so t2 = 1, thus t.t = 1. If we differentiate this, we get that t.t‘ = 0, which specifies a direction normal to the curve, provided t‘ is not equal to zero. This is because if the dot product of two vectors is zero, then those two vectors are perpendicular to each other.

Let us define t’ = Kwhere the unit vector n(s) is called the principal normal and K(s) is called the curvature. Note that we can always make K positive by choosing an appropriate direction for n.

Another interesting quantity is the radius of curvature, a, which is given by

a = 1/curvature

Now that we have n and t we can define a new vector x n, which is orthonormal to both t and n. This is called the binormal. Using this, we can then examine the torsion of the curve, which is given by

T(s) = –b’.n

Image result for binormal tangent

Intrinsic Geometry

As the plane is rotated about n we can find a range

Screen Shot 2017-02-14 at 8.16.01 AM.pngwhereScreen Shot 2017-02-14 at 8.16.08 AM.png and Screen Shot 2017-02-14 at 8.16.14 AM.png are the principal curvatures. Then

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is called the Gaussian curvature.

Gauss’ Theorema Egregium (which literally translates to ‘Remarkable Theorem’!) says that K is intrinsic to the surface. This means that it can be expressed in terms of lengths, angles, etc. which are measured entirely on the surface!

For example, consider a geodesic triangle on a surface S.

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Let θ1, θ2, θ3 be the interior angles. Then the Gauss-Bonnet theorem tells us that

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which generalises the angle sum of a triangle to curved space.

Let us check this when S is a sphere of radius a, for which the geodesics are great circles. We can see that Screen Shot 2017-02-14 at 8.16.08 AM.png=Screen Shot 2017-02-14 at 8.16.14 AM.png= 1/a, and so K = 1/a2, a constant. As shown below, we have a family of geodesic triangles D with θ1 = α, θ2 = θ3 = π/2.

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Since K is constant over S,

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Then θ1 + θ2 + θ3 = π + α, agreeing with the prediction of the theorem.

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VIDEO: Banach–Tarski Paradox

The Banach-Tarski Paradox is a theorem in geometry which states that:

“It is possible to decompose a ball into five pieces which can be reassembled by rigid motions to form two balls of the same size as the original.”

It was first stated in 1924, and is called a paradox as it contradicts basic geometric intuition.

An alternate version of this theorem tells us that:

“It is possible to take a solid ball the size of a pea, and by cutting it into a finite number of pieces, reassemble it to form a solid ball the size of the sun.”

Below is an awesome video explaining how this paradox works:

 

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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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symmetries

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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