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 | 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 | Source: Wikipedia
Surface Area: 16r2
Volume: , 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 .
Then the bicylinder volume is:
Tricylinder | Source: Wikipedia
A heptadecagon (or a 17-gon) is a seventeen sided polygon.
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 in terms of square roots, which Gauss gave in his book Disquistiones Arithmeticae:
An explicit construction was given by Herbert Willian Richmond in 1893.
A ford circle is a circle with centre , and radius where p and q are coprime integers.
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.
Let s be the sum of the radii:
However, we have that and so , thus the distance between circles is greater or equal to the sum of the radii of the circles. There is equality iff
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:
is less than 1.
From the definition, the area is
Simplifying this expression gives us
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
Proof of the identity
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.
From ‘Proof without words‘ by Roger Nelsen
By Sidney H. Kung
By Shirley Wakin
Previous ‘Proof Without Words‘: Part 1 | Part 2
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 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.
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).
Dini’s surface is given by the following parametric equations:
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).
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 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 | 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.
- A nine-point circle bisects a line segment going from the corresponding triangle’s orthocenter to any point on its circumcircle.
- 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!