Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
A review of the website http://www.bangor.ac.uk/cpm/exhib/
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots, prime knots, crossing numbers and knot arithmetic.
As an adventure in pure mathematics, and a good science fiction
story, it is hard to beat Edwin Abbott's Flatland, a classic from
the 1880's. This strange space is peopled by geometrical figures
that think and speak and have human emotions. The book explores the
perceptions these beings have of higher dimensions. Equally
fanciful, these Earthshapes embody some mathematics and take us
into other science fiction worlds.
The Earthshapes were dreamed up by Joseph Portney about thirty
years ago when flying over the North pole on board a U.S. Air Force
KC-135. Joe looked on the icy terrain below and mused to himself,
"What if the Earth were" and these odd shaped models were the
result. Joe graduated from the US Naval Academy, served as a
navigator bombardier in the US Air Force and since then has worked
for many years for Litton Guidance & Control Systems on
navigation problems concerned with the safe guidance and control of
high altitude, long range, aircraft (both military and commercial).
By the application of mathematics to re-programming the guidance
system's computer, a plane could be guided safely over and around
the odd curves and corners of these hypothetical worlds. You will
find a lot more fascinating information on the subject and some
'brain stretchers' on the Litton website . There you will find
Portney's Ponderables which feature the Earthshapes and also Time
Tunnel ( replication of Eratosthenes' ancient measurement of the
circumference of the Earth), Brain Games (a bi-monthly science
puzzle) and Other Worlds (a link to the UK British Museum ).
Reproduced by permission of Joseph Portney. Copyright 1998
Litton Systems, Inc. All rights reserved.
Here are some questions for class discussion:
Of course these shapes belong to a fantasy. However, determining
the shape of the Earth has long been a difficult problem and a
controversial matter. In the Bible the Earth was said to have 4
corners, and in another passage to be circular. Some of the ancient
Greeks believed the Earth to be spherical, others that it was flat.
About 230 B.C. Eratosthenes calculated the Earth's
circumference and he was the first to attempt to produce a map
of the World based on a system of lines of latitude and
longitude. However the belief that the Earth was flat persisted
through to the middle ages, to be disproved by the voyages of
Columbus, Magellan and others.
Because of surface curvature, mapping the Earth has posed a
problem, it cannot be cut and laid out on a flat surface without
distortion. Maps which distort angles are useless for navigation.
The mathematician Geradus Mercator devised a map, first
used in 1569, which allowed mariners to measure angles directly
from the map as the angle between any two directions on the
Earth's surface was accurately re-produced. The map is drawn
inside a tight fitting cylinder which touches the Earth around
the equator and has its axis through the poles. Points where the
rays from this central axis and parallel to the equator cut the
Earth's surface are mapped to corresponding points on the
cylinder. This mapping stretches the lines of latitude to make
the meridians parallel and greatly distorts the land near the
Nowadays satellite tracking is used to determine the Earth's
shape improving the accuracy of maps and the Earth is considered to
be slightly 'pear shape and not exactly spherical.