Jackson an’ Johnson / Murphy an’ Bronson /
One by one dey come / An’ one by one to dreamland dey go.
[From Carmen Jones. Lyrics: Oscar Hammerstein]
Euclid’s Theorem I-47, also known as Pythagoras’ Theorem, in Oliver Byrne’s colourful text, The Elements of Euclid.
Two young high-school students from New Orleans, Ne’Kiya Jackson and Calcea Johnson, recently presented a new proof of the Pythagorean theorem at a meeting of the American Mathematical Society in Georgia. It has been widely believed that no proof based on trigonometry was possible but we now know that to be false. Continue reading ‘The Waffle Cone and a new Proof of Pythagoras’ Theorem’
in reduced form (
and 
coprime), a Ford circle is a circle with center at 
and radius 
. There is a Ford circle associated with every rational number. Every Ford circle is tangent to the horizontal axis and each two Ford circles are either tangent or disjoint from each other.
are convenient. Here 
is the longitude and 
the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967):
. Another triangle, with sides 
and 
is stacked upon the first. This process continues, giving hypotenuse lengths 
for all 
. 
. He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.
is the angle through which the disk has rotated. The centre of the disk is at 
.
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 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.
characterizes the intrinsic geometry of a surface.
It was a great shock to the Pythagoreans to discover that the diagonal of a unit square could not be expressed as a ratio of whole numbers. This discovery represented a fundamental fracture between the mathematical domains of Arithmetic and Geometry: since the Greeks recognized only whole numbers and ratios of whole numbers, the result meant that there was no number to describe the diagonal of a unit square.
where 
is the radius. Some of us may also remember that the volume of a ball — the interior of a sphere — is 
.
, the length of the major axis. The gardener puts down two stakes and loops a piece of rope around them. Using a stick, he pulls the loop taut, marking the points around a curve. This is illustrated here.
Ireland has four provinces, the principal city in each being the provincial capital: Belfast, Cork, Dublin and Galway. The map here shows the location of these cities. Now imagine a company that needs to visit and to deliver goods frequently to all four cities. Where might they locate their HQ to minimize transport costs and travel times?
is the position of the HQ and 
are the positions of the cities.
Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times [see 
for 
is rotated in 3-space about the x-axis.
ThatsMaths 