Quadrivium: The Noble Fourfold Way

According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato’s Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD [see TM119 or search for “thatsmaths” at irishtimes.com].

Quadrivium-Book

Image from here.

It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.

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Inertial Oscillations and Phugoid Flight

The English aviation pioneer Frederick Lanchester (1868–1946) introduced many important contributions to aerodynamics. He analysed the motion of an aircraft under various consitions of lift and drag. He introduced the term “phugoid” to describe aircraft motion in which the aircraft alternately climbs and descends, varying about straight and level flight. This is one of the basic modes of aircraft dynamics, and is clearly illustrated by the flight of gliders.

Glider-Loop-20

Glider in phugoid loop [photograph by Dave Jones on website of Dave Harrison]

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Robert Murphy, a “Brilliant Meteor”

A brilliant meteor that flared intensely but all too briefly”; this was how Des MacHale described the Cork-born mathematician Robert Murphy in his biography of George Boole, first professor of mathematics in Cork. Murphy was a strong influence on Boole, who quoted liberally from his publications [see TM118 or search for “thatsmaths” at irishtimes.com].

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Patterns in Poetry, Music and Morse Code

Suppose we have to ascent a flight of stairs and can take only one or two steps at a time. How many different patterns of ascent are there? We start with the simplest cases. With one step there is only one way; with two, there are two: take two single steps or one double step. With three steps, there are three possibilities. We can now proceed in an inductive manner.

Staircase-01

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The Beer Mat Game

Alice and Bob, are enjoying a drink together. Sitting in a bar-room, they take turns placing beer mats on the table. The only rules of the game are that the mats must not overlap or overhang the edge of the table. The winner is the player who puts down the final mat. Is there a winning strategy for Alice or for Bob?

Beermats-picture

Image from Flickr. 

We start with the simple case of a circular table and circular mats. In this case, there is a winning strategy for the first player. Before reading on, can you see what it is?

* * *

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Fractal Complexity of Finnegans Wake

Tomorrow we celebrate Bloomsday, the day of action in Ulysses. Most of us regard Joyce’s singular book as a masterpiece, even if we have not read it. In contrast, Finnegans Wake is considered by some as a work of exceptional genius, by others as impenetrable bafflegab [See TM117 or search for “thatsmaths” at irishtimes.com].

FW-Fractals-03-Squeezed.jpg

Wavelet transform of sentence length sequence in Ulysses. Note the structural change around sentence number 13,000. Image from Drozdz, et al (2016).

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A Remarkable Pair of Sequences

The terms of the two integer sequences below are equal for all {n} such that {1<n<777{,}451{,}915{,}729{,}368},  but equality is violated for this enormous value and, intermittently, for larger values of {n}.

TicTacToe-2-Sequences

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