The cycloid, the locus of a point on the rim of a rolling disk.
The Cycloid
The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (see figure). The parametric equations for the cycloid are
where 
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Continue reading ‘Mamikon’s Theorem and the area under a cycloid arch’
be the radius of the circle and 
the distance from the axis to the centre of the circle, with 
.
as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result
. The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,
can be found using derangement numbers. Recall from last week’s post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position.
. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya Conjecture:
are the
, and ratios of these, the positive rational numbers 
. It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.
, which include rationals, irrationals like 
and transcendental numbers like 
.
of points in the plane 
(Fig. 1a). How large a circle is required to enclose them. More specifically, what is the minimum radius of such a bounding circle? The answer is given by Jung’s Theorem.
for s=1.1 (red), s=1.01 (blue) and s=1.001 (black).
we can usually find 
, whereas we may not be able to find 
.
. The number of atoms in the universe is estimated to be about 
. When we consider permutations of large sets, even more breadth-taking numbers emerge.
changed by one part in a million.
. Although the angular momentum did not change, the rotation axis moved in the body frame. The nut continued to flip back and forth, although there were no forces or torques acting on it.
, 
, 
using the integral
the amplitude and wrote 
. It can be difficult to understand what motivated his definitions. We will define the elliptic functions 
for which the function is zero. A simple iterative method was devised by Isaac Newton and refined by Joseph Raphson. It is known either as Newton’s method or as the Newton-Raphson method. It usually produces highly accurate approximations to the roots of the equation 
.
“
For many decades, a search has been under way to find a theory of everything, that accounts for all the fundamental physical forces, including gravity. The dictum “physics is geometry” is a guiding principle of modern theoretical physics. Einstein’s General Theory of Relativity, which emerged just one hundred years ago, is a crowning example of this synergy. He showed how matter distorts the geometry of space and this geometry determines the motion of matter. The central idea is encapsulated in an epigram of John A Wheeler:
From cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved chasing and killing animals, and success depended on an understanding of relative speeds and optimal pursuit paths.
