Posts Tagged 'Geometry'

The Miraculous Spiral on Booterstown Strand

We all know what a spiral looks like. Or do we? Ask your friends to describe one and they will probably trace out the form of a winding staircase. But that is actually a helix, a curve in three-dimensional space. A spiral is confined to a plane – it is a flat curve. In general terms, a spiral is formed by a point moving around a fixed centre while its distance increases or decreases as it revolves [see TM145, or search for “thatsmaths” at].


The spiral sandbank on Booterstown strand (satellite image digitally enhanced by Andrew Lynch).

Continue reading ‘The Miraculous Spiral on Booterstown Strand’

Optical Refinements at the Parthenon

The Parthenon is a masterpiece of symmetry and proportion. This temple to the Goddess Athena was built with pure white marble quarried at Pentelikon, about 20km from Athens. It was erected without mortar or cement, the stones being carved to great accuracy and locked together by iron clamps. The building and sculptures were completed in just 15 years, between 447 and 432 BC. [TM141 or search for “thatsmaths” at].


Continue reading ‘Optical Refinements at the Parthenon’

A Glowing Geometric Proof that Root-2 is Irrational

Tennenbaum-00It 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.

Continue reading ‘A Glowing Geometric Proof that Root-2 is Irrational’

Marden’s Marvel

Although polynomial equations have been studied for centuries, even millennia, surprising new results continue to emerge. Marden’s Theorem, published in 1945, is one such — delightful — result.


Cubic with roots at x=1, x=2 and x=3.

Continue reading ‘Marden’s Marvel’

Geodesics on the Spheroidal Earth-II

Geodesy is the study of the shape and size of the Earth, and of variations in its gravitational field. The Earth was originally believed to be flat, but many clues, such as the manner in which ships appear and disappear at the horizon, and the changed perspective from an elevated vantage point, as well as astronomical phenomena, convinced savants of its spherical shape. In the third century BC, Eratosthenes accurately estimated the circumference of the Earth [TM137 or search for “thatsmaths” at].


Geodesic at bearing of 60 degrees from Singapore. Passes close to Quito, Ecuador. Note that it is not a closed curve: it does not return to Singapore.

Continue reading ‘Geodesics on the Spheroidal Earth-II’

Geodesics on the Spheroidal Earth – I

Both Quito in Ecuador and Singapore are on the Equator. One can fly due eastward from Singapore and reach Quito in due course. However, this is not the shortest route. The equatorial trans-Pacific route from Singapore to Quito is not a geodesic on Earth! Why not?


A drastically flattened spheroid. Clearly, the equatorial route between the blue and red points is not the shortest path.

Continue reading ‘Geodesics on the Spheroidal Earth – I’

The Evolute: Envelope of Normals

Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute.


Sin t (blue) and its evolute (red).

Continue reading ‘The Evolute: Envelope of Normals’

Last 50 Posts