Plenaire lezingen

Pomerance, C. (ed.). (1990). Cryptology and computational number theory. Proceedings of Symposium Applications. Mathematics 42, American Mathematical Society.
Lenstra, A.K. en H.W. Lenstra, Jr. (eds). (1993). The development of the number field sieve, Lecture Notes in Math. 1554. Springer-Verlag.

 

From figure to form - a geometrical excursion into nature

Vagn Lundsgaard Hansen
Mathematical Institute, Technical University of Denmark, Lygnby

The Greeks were convinced that the universe is structured according to mathematical laws. This lecture will focus on geometrical aspects of this philosophy and in particular on the interplay between the concrete and the abstract side of geometry.
Cosmic forms: From the planets' movements to the shape of an ellipse
We draw a line from the attempts in antiquity to model the movements of the planets, over Kepler and Copernicus to the shapes of the conics.
Polyhedra: From geometry to topology
We illustrate quantitative and qualitative aspects of geometry in the world of polyhedra, where Euler's theorem marks the step from geometry to topology.
Ornamental forms: From decorations to mathematical patterns
We examine the geometry of tilings of plane areas with regular polygons to demonstrate the limitations on the symmetries in regular patterns in the plane.
Optimal forms: From intuition to mathematical proof
By way of the isoperimetric problem we show the need for mathematical proofs.
Non-Euclidean geometry: From postulate to axiom
Around 1830 it turned out that the parallel postulate in Euclid's Elements cannot be proved; it has to be added as an axiom characterizing Euclidean geometry. We describe Poincaré's model of a non-Euclidean geometry: the hyperbolic plane.
Hyperbolic shapes: Tilings of the hyperbolic plane
Tilings of the hyperbolic plane are far richer than tilings of the Euclidean plane. We shall explain why.
Riemannian shapes: Geometry and topology of surfaces
As a topological object, a closed surface in 3-space without boundary edges is a sphere with a number (the genus of the surface) of handles attached. We shall explain how the geometry of the surface depends on the genus.
Curved shapes: From soap films to minimal surfaces
We take a look at curved shapes representable by soap films.
Forms in nature: The great book of geometry
As an example of geometry in nature, we describe the mathematics of the shell of a snail.

Literatuur en bronnen
Hansen, V. L. (1993). Geometry in Nature. Massachusetts, USA: K. Peters Ltd. Wellesley.
Hansen, V. L. (1994). The Magic World of Geometry - I. The Isoperimetric Problem. Elemente der Mathematik, Vol. 49, No. 2, 61-65.

 

Gyroscopes and boomerangs

Sir Christopher Zeeman
Hertford College, Oxford, United Kingdom

This lecture is taken from material that the lecturer has been teaching in Mathematics Masterclasses to gifted 13 year-olds during the last 15 years.

Starting from a form of Newton's law the lecture will develop the theory of gyroscopes, both qualitative and quantitative. Since 13 year-olds will not have done calculus and relatively little algebra, the development uses no calculus and not much algebra. It will explain why gyroscopes process and in which direction they process, by proving the gyro law: the spin axis chases the torque axis. The lecture will then prove a quantitative formula for predicting the rate of procession of a spinning bicycle wheel, and will confirm the predictions by experiment. The experiments are simple and robust, using only home made equipment, and are therefore very suitable for classroom use.
The lecture will apply the theory with demonstrations to spinning tops, to explain why they go to sleep and then suddenly wake up. The lecture will then describe how to make and throw a boomerang, and will explain why a boomerang returns because it is both a gyroscope and an aircraft. Finally the lecturer will attempt to throw and catch a boomerang.

 

 

Literatuur en bronnen
The lecturer has made a video, also entitled `Gyroscopes and boomerangs', with an accompanying book containing notes, worksheets, solutions and appendices. The video lasts one hour, and is divided into three sections, between which the viewer is invited to tackle the worksheets. The video can be used either with teacher supervision in the classroom or by students studying alone. The appendices contain follow-up material including an introduction to calculus, the quantitative theory of boomerangs, and a rigorous proof of why a spinning hard-boiled egg will stand up on end.

The video and book can be purchased from the Royal Institution (attention Ms. Cripps), 21 Albemarle Street, London W1X 4BS, England. A copy will be on show at the meeting.

 

De betrekking tussen wiskunde en mijn kunst

Peter Struycken
Gorinchem

Wiskunde, in mijn geval een eenvoudige vorm van logica, bepaalt mijn beeldend werk. Voor mijn werk, met veranderende kleurverhoudingen als onderwerp, gebruik ik een computer. De rekenvoorschriften die ik bedenk - het geheel van voorwaarden en functies - bepalen de aard en het bereik van de visuele uitkomsten. Met deze rekenvoorschriften, die ik een functie laat zijn van tijd en ruimte, kunnen kleurbeelden van onbegrensde afmeting en detail onbeperkt in tijd veranderen. Maar nooit zal zelfs maar een fractie van een beeld op enigerlei tijdstip berekend worden dat niet volstrekt bepaald wordt door de logische vorm van de voorschriften.

Door de rekenvoorschriften te wijzigen kan ik andere beelden berekenen die, eveneens van onbegrensde afmeting, onbeperkt in tijd veranderen. Nooit zullen echter deze beelden overeenkomen met die uit eerdere voorschriften. Ieder voorschrift opent en begrenst een naar aard en bereik gescheiden visuele wereld die onbeperkt uitgebreid is.
En daarmee ligt ook de kwaliteit van de beelden vast. Die kwaliteit kan als goed of slecht beoordeeld worden door kenners van kunst, maar zij wordt iedere keer opnieuw en onveranderlijk door de aard en het bereik bepaald die de logische vorm van de voorschriften veroorzaakt op kleurverhoudingen in tijd en ruimte.

Alhoewel met bovenstaande de rol van de wiskunde voor aard, omvang en kwaliteit van mijn werk is aangegeven, blijft de vraag naar de zin om op deze gestructureerde manier kunst te maken onbeantwoord. Die vraag kan alleen vanuit de kunst beantwoord worden, waarbij de wiskunde geen rol speelt.