Nail Supremacy

Ὁ γαρ ἡδονής και ἀλγηδόνος ἧλος, ὃς πρὸς το σώμα τήν ψυχην προσηλοῖ, μέγιστον κακὸν ἔχειν ἔοικε, τὸ τα αἰσθητά ποιεῖν ἐναργέστερα τῶν νοητῶν, καὶ καταβιάζεσθαι καὶ πάθει μᾶλλον ἢ λόγῳ κρίνειν τήν διάνοιαν.

• ΠΡΟΒΛΗΜΑ Β’. Πώς Πλάτων ἔλεγε τον θεὸν άεὶ γεωμετρεῖν.

Nam voluptatis et doloris ille clavus, quo animus corpori affigitur, id videtur maximum habere malum, quod sensilia facit intelligibilibus evidentiora, vimque facit intellectui, ut affectionem magis quam rationem in judicando sequatur.

• QUÆSTIO II: Qua ratione Plato dixerit, Deum semper geometriam tractare.

For the nail of pain and pleasure, which fastens the soul to the body, seems to do us the greatest mischief, by making sensible things more powerful over us than intelligible, and by forcing the understanding to determine them rather by passion than by reason.

• Plutarch’s Symposiacs, QUESTION II: What is Plato’s Meaning, When He Says that God Always Plays the Geometer?

Know Your Limaçons

Front cover of The Penguin Dictionary of Curious and Interesting Geometry by David WellsThe Penguin Dictionary of Curious and Interesting Geometry, David Wells (1991)

Mathematics is an ocean in which a child can paddle and an elephant can swim. Or a whale, indeed. This book, a sequel to Wells’ excellent Penguin Dictionary of Curious and Interesting Mathematics, is suitable for both paddlers and plungers. Plumbers, even, because you can dive into some very deep mathematics here.

Far too deep for me, I have to admit, but I can wade a little way into the shallows and enjoy looking further out at what I don’t understand, because the advantage of geometry over number theory is that it can appeal to the eye even when it baffles the brain. If this book is more expensive than its prequel, that’s because it needs to be. It’s a paperback, but a large one, to accommodate the illustrations.

Fortunately, plenty of them appeal to the eye without baffling the brain, like the absurdly simple yet mindstretching Koch snowflake. Take a triangle and divide each side into thirds. Erect another triangle on each middle third. Take each new line of the shape and do the same: divide into thirds, erect another triangle on the middle third. Then repeat. And repeat. For ever.

A Koch snowflake (from Wikipedia)

A Koch snowflake (from Wikipedia)

The result is a shape with a finite area enclosed by an infinite perimeter, and it is in fact a very early example of a fractal. Early in this case means it was invented in 1907, but many of the other beautiful shapes and theorems in this book stretch back much further: through Étienne Pascal and his oddly organic limaçon (which looks like a kidney) to the ancient Greeks and beyond. Some, on the other hand, are very modern, and this book was out-of-date on the day it was printed. Despite the thousands of years devoted by mathematicians to shapes and the relationship between them, new discoveries are being made all the time. Knots have probably been tied by human beings for as long as human beings have existed, but we’ve only now started to classify them properly and even find new uses for them in biology and physics.

Which is not to say knots are not included here, because they are. But even the older geometry Wells looks at would be enough to keep amateur and recreational mathematicians happy for years, proving, re-creating, and generalizing as they work their way through variations on all manner of trigonomic, topological, and tessellatory themes.

Previously pre-posted (please peruse):

Poulet’s Propeller — discussion of Wells’ Penguin Dictionary of Curious and Interesting Numbers (1986)