Influential Mathematicians: Gauss (3)

Probability and Statistics

Gauss introduced what is now known as the Gaussian distribution: he showed how probability can be represented by a bell-shaped curve, with peaks around the mean when falls off quickly towards plus or minus infinity.

File:Normal Distribution PDF.svg
Source: Wikipedia

He also created the Gaussian function: a function of the form

{\displaystyle f(x)=ae^{-{\frac {(x-b)^{2}}{2c^{2}}}}}

for arbitrary real constants a, b and c.

Modular Arithmetic

The modern approach to modular arithmetic was developed by Gauss in his book Disquisitiones Arithmeticae, published in 1801.  This now has application in number theory, abstract algebra, computer science, cryptography, and even in visual and musical art.

Surveying

Whilst doing a surveying job for the Royal House of Hanover in the years after 1818, Gauss was also looking into the shape of the Earth and started to question what the shape of space itself was. This led him to question Euclidean geometry – one of the central tenets of the whole mathematics, which premised a flat universe, rather than a curved one. He later claimed that as early as 1800 he had already started to consider types of non-Euclidean geometry (where the parallel axiom does not hold), which were consistent and free of contradiction. However, to avoid controversy, he did not publish anything in this area and left the field open to Bolyai and Lobachevsky, although he is still considered by some to be the pioneer of non-Euclidean geometry.

This survey work also fuelled Gauss’ interest in differential geometry, which uses differential calculus to study problems in geometry involving curves and surfaces. He developed what has become known as Gaussian curvature. This is an intrinsic measure of curvature that depends only on how distances are measured on the surface, not on the way it is embedded in space.

Positive, negative and zero Gaussian curvature of a shell
Source: shellbuckling.com

His achievements during these years, however, was not only limited to pure mathematics. He invented the heliotrope, which is an instrument that uses a mirror to reflect sunlight over great distances to mark positions in a land survey.

Image result for heliotrope gauss
Heliotrope | Source: Wikipedia

All in all, this period of time was one of the most fruitful periods of his academic life; he published over 70 papers between 1820 and 1830.

In later years, he worked with Wilhelm Weber to make measurements of the Earth’s magnetic field, and invented the first electric telegraph.

Read part 1 here and part 2 here.

Let me know what you think of this new series! M x

 

Influential Mathematicians: Gauss (1)

I decided to start a new series on influential mathematicians, starting with Gauss, one of my personal favourites. Carl Friedrich Gauss (1777-1855) is considered to be one of the greatest mathematicians in the 19th century, and is sometimes referred to as the “Prince of Mathematics”.

His discoveries influenced and left a lasting mark in a variety of different areas, including number theory, astronomy, geodesy, and physics, particularly the study of electromagnetism.

Born in Brunswick, Germany to poor, working class parents, he was discouraged from attending school from his father, a gardner and brick-layer, who expected Gauss would follow one of the family trades. However, Gauss’ mother and uncle recognised Gauss’ early genius and knew he must develop this gift with a proper education.

In arithmetic class, at the age of 10, Gauss showed his skills as a maths prodigy. A well known anecdote about Gauss and his early school education is about when the strict schoolmaster gave the following assignment:

“Write down all the whole numbers from 1 to 100 and add up their sum.”

They expected this assignment to take a while to complete but after a few seconds, to the teacher’s surprise, Carl placed his slate on the desk in front of the teacher, showing he was done with the question. His other classmates took a much longer time to complete the assignment. At the end of class time, although most other students answers were wrong, Gauss’ was correct: 5050. Carl then explained to the teacher that he found the result as he could see that 1+100 = 101, 2+99=101, etc. So he could find 50 pairs of numbers that each add up to 101, and so 50*101 = 5050. I don’t know about you but I definitely could not come up with this sort of argument at the age of 10…

Although his family was poor, Gauss’ intellectual abilities drew the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at the age of 15, and then to the University of Göttingen – a very prestigious university – where he stayed from 1795 to 1798. During this period, Gauss discovered many important theorems.

Prime Numbers

No pattern had previously been found in the occurrence of prime numbers until Gauss. Although the occurrence of the primes seems to be completely random, by approaching the problem from a different angle and graphing the incidence of primes as the numbers increased, he noticed a rough trend: as numbers increased by 10, the probability of the numbers reduced by a factor of around 2. However, as his method only gave him an approximation, and as he could not definitively prove his findings, he kept them a secret until much later in his life.

Graphs of the density of prime numbers

1796

1796 is known as Gauss’ “annus mirabilis” (means “wonderful year” and is used to refer to several years during which events of major importance are remembered).  In 1796:

  • Gauss constructed a regular 17-sided heptadecagon, which had previously been unknown, using only a ruler and a compass. This was a major advance in geometry since the time of the Greeks.
  • Gauss formulated this prime number theorem on the distribution of prime numbers among the integers, which states that \displaystyle \lim_{n\rightarrow\infty}\left[ \frac{\pi(n)}{n/\log n} \right] = 1 \,.  Here {\pi(n)} is the number of primes less than or equal to n. We can also write {\pi(n) \sim n/\log n}.
  •  Gauss proved that every positive integer can be represented as the sum of at most 3 triangular number

More about Gauss in the next post!

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MATHS BITE: Steinmetz Solid

Named after mathematician Charles Proteus Steinmetz, a Steinmetz solid is the solid body generated by the intersection of two or three cylinders with the same radii at right angles.

Bicylinder_Steinmetz_solid
Bicylinder | Source: Wikipedia

These solids are named after Steinmetz as he managed to determine the volume of the intersection, though these solids were known long before he studied them.

If two/three cylinders intersect then the intersection is called a bicylinder/tricylinder.

Bicylinder

File:Steinmetz-solid.svg
Bicylinder | Source: Wikipedia

Surface Area: 16r2

Volume: {\frac  {16}{3}}r^{3}, where r is the radius of both cylinders.

The volume of the cube minus the volume of the eight pyramids is the volume of the bicylinder: the volume of 8 pyramids is \textstyle 8\times {\frac  {1}{3}}r^{2}\times r={\frac  {8}{3}}r^{3}.

 Then the bicylinder volume is:

\textstyle (2r)^{3}-{\frac  {8}{3}}r^{3}={\frac  {16}{3}}r^{3}

Tricylinder

Surface Area: 3(16-8{\sqrt  {2}})r^{2}.\,

Volume: (16-8{\sqrt  {2}})r^{3}\,

Tricylinder Steinmetz solid.gif
Tricylinder | Source: Wikipedia

 

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MATHS BITE: Heptadecagon

A heptadecagon (or a 17-gon) is a seventeen sided polygon.

File:Regular polygon 17 annotated.svg
Regular Heptadecagon | Wikipedia

Constructing the Heptadecagon

In 1796, Gauss proved, at the age of 19 (let that sink in…) that the heptadecagon is constructible with a compass and a straightedge, such as a ruler. His proof of the constructibility of an n-gon relies on two things:

  • the fact that “constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions“;
  • the odd prime factors of n are distinct Fermat primes.

Constructing the regular heptadecagon involves finding the expression for the cosine of  2\pi /17 in terms of square roots, which Gauss gave in his book Disquistiones Arithmeticae:

{\displaystyle {\begin{aligned}16\,\cos {\frac {2\pi }{17}}=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}\\=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}.\end{aligned}}}
Source: Wikipdia

An explicit construction was given by Herbert Willian Richmond in 1893.

Regular Heptadecagon Using Carlyle Circle.gif
Source: Wikipedia

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MATHS BITE: Ford Circles

A ford circle is a circle with centre (p/q,1/(2q^{2})), and radius 1/(2q^{2}), where p and q are coprime integers.

File:Ford circles colour.svg
Source: Wikipedia

Notice that each Ford Circle is tangent to to the horizontal axis and any two Ford circles are either tangent or disjoint. The latter statement can be proven by finding the squared distance d^2 between the centres of the circles with (p,q) and (p’,q’) as the pairs of coprime integers.

latex-image-1.png

Let s be the sum of the radii:

latex-image-2.png

Then

latex-image-3.png

However, we have that latex-image-4.png and so latex-image-5.png, thus the distance between circles is greater or equal to the sum of the radii of the circles. There is equality iff

latex-image-6.png

In this case, the circles are tangent to one another.

Total area of Ford Circles

(taken from Wikipedia)

As no two ford circles intersect, it follows immediately that the total area of the Ford circles:

\left\{C[p,q]:0\leq {\frac  {p}{q}}\leq 1\right\}

is less than 1.

From the definition, the area is

A=\sum _{{q\geq 1}}\sum _{{(p,q)=1 \atop 1\leq p<q}}\pi \left({\frac  {1}{2q^{2}}}\right)^{2}.

Simplifying this expression gives us

A={\frac  {\pi }{4}}\sum _{{q\geq 1}}{\frac  {1}{q^{4}}}\sum _{{(p,q)=1 \atop 1\leq p<q}}1={\frac  {\pi }{4}}\sum _{{q\geq 1}}{\frac  {\varphi (q)}{q^{4}}}={\frac  {\pi }{4}}{\frac  {\zeta (3)}{\zeta (4)}},

noting that the last equality is given by considering the Dirichlet generating function for Euler’s totient function φ(q).

Given that ζ(4) = π^4/90, we get

A={\frac  {45}{2}}{\frac  {\zeta (3)}{\pi ^{3}}}\approx 0.872284041.

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Proof Without Words #3

Proof of the identity

Screen Shot 2017-07-30 at 5.34.22 PM.png

HCfGOYp.gif


dkoCZ.png

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.

g4n8SZX.jpg



From ‘Proof without words‘ by Roger Nelsen

Screen Shot 2017-07-30 at 5.47.56 PM.png
By Sidney H. Kung
Screen Shot 2017-07-30 at 5.48.12 PM.png
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.
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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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

Screen Shot 2017-02-14 at 8.17.34 AM.png

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.

Screen Shot 2017-02-14 at 8.20.37 AM.png

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

Screen Shot 2017-02-14 at 8.22.10 AM.png

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.

screen-shot-2017-02-14-at-8-26-15-am

Since K is constant over S,

Screen Shot 2017-02-14 at 8.27.09 AM.png

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

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