Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].
characterizes the intrinsic geometry of a surface.
Continue reading ‘Gaussian Curvature: the Theorema Egregium‘
Posts Tagged 'Geometry'
Gaussian Curvature: the Theorema Egregium
Published December 27, 2018 Occasional Leave a CommentTags: Gauss, Geometry
characterizes the intrinsic geometry of a surface.
A Trapezoidal Prism on the Serpentine
Published September 13, 2018 Occasional Leave a CommentTags: Geometry
Walking in Hyde Park recently, I spied what appeared to be a huge red pyramid in the middle of the Serpentine. On closer approach, and with a changing angle of view, it became clear that it was prismatic in shape, composed of numerous barrels in red, blue and purple.
Changing perspective on approach to the Mastaba
The Miraculous Spiral on Booterstown Strand
Published August 16, 2018 Irish Times Leave a CommentTags: biology, Geometry, Geophysics
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 irishtimes.com].
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
Published June 21, 2018 Irish Times Leave a CommentTags: Geometry, History
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 irishtimes.com].
A Glowing Geometric Proof that Root-2 is Irrational
Published May 24, 2018 Occasional 1 CommentTags: Geometry
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.
Continue reading ‘A Glowing Geometric Proof that Root-2 is Irrational’
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.
Geodesics on the Spheroidal Earth-II
Published April 19, 2018 Irish Times Leave a CommentTags: Geometry, Geophysics, Navigation, Pythagoras
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 irishtimes.com].
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.
