William Thurston
(October 30, 1946 – August 21, 2012)
Using your imagination…
How do you imagine geometric figures in your head?
“How many edges does a cube have?”
Exercises in imagining
On your tables there are some cards.
Find another person to work with in going through these images.
Evoke the images by talking about them, not by drawing them. It
will probably help to close your eyes, although sometimes
gestures and drawings in the air will help.
Skip around to try to find exercises that are the right level for you.
Take a 3 x 4 rectangular array of dots in
the plane, and connect the dots
vertically and horizontally. How many
squares are enclosed?
Cut off each corner
of a square, as far as
the midpoints of the
edges. What shape
is left over? How can
you re-assemble the
four corners to make
another square?
Mark the sides of
an equilateral
triangle into
thirds. Cut off
each corner of
the triangle, as
far as the marks.
What do you get?
How many different
colours are required
to colour the faces of
a cube so that no
two adjacent faces
have the same
colour?
Rest a tetrahedron on
its base, and cut it
halfway up. What
shape is the smaller
piece? What shapes
are the faces of the
larger pieces?
Mark the sides of a
square into thirds,
and cut off each of
its corners back to
the marks. What
does it look like?
Take two squares.
Place the second
square centred over the
first square but at a
forty-five degree angle.
What is the intersection
of the two squares?
Find a path through edges of
the dodecahedron which visits
each vertex exactly once.
What three-dimensional
solid has circular profile
viewed from above, a
square profile viewed from
the front, and a triangular
profile viewed from the side?
Do these three profiles
determine the threedimensional shape?
How many colours are
required to colour the
faces of an octahedron
so that faces which
share an edge have
different colours?
This was part of a course that aimed to convey the richness, diversity,
connectedness, depth and pleasure of mathematics. The title is taken from the
classic book by Hilbert and Cohn-Vossen, `Geometry and the Imagination'.
One aim of the course is to develop the imagination. Imagination is an essential
part of mathematics, not only the facility which is imaginative, but also the facility
which calls to mind and manipulates mental images.
“The spirit of mathematics is not captured by spending 3 hours solving 20 lookalike homework problems. Mathematics is thinking, comparing, analysing,
inventing, and understanding. The main point is not quantity or speed-the main
point is quality of thought.”
Bill Thurston - the mathematician
Thurston’s early work was in foliations - a way to see a how a
geometric object can be pieced together.
Thurston’s later work was on the building blocks for 3-manifolds
and the importance of Hyperbolic geometry.
Thurston was awarded the Fields Medal in 1982 for his
contributions to the study of 3-manifolds.
In 2005 Thurston won the first AMS Book Prize, for Threedimensional Geometry and Topology. The prize "recognises an
outstanding research book that makes a seminal contribution to
the research literature"
Could the difficulty in giving a good direct definition of mathematics
be an essential one, indicating that mathematics has an essential
recursive quality?
Along these lines we might say that mathematics is the smallest
subject satisfying the following:
• Mathematics includes the natural numbers and plane and solid
geometry.
• Mathematics is that which mathematicians study.
• Mathematicians are those humans who advance human
understanding of mathematics.
Curvature… different geometries
How can we build up the
surface of pieces of fruit?
banana
pear
apple
orange
Building a surface
http://nrich.maths.org/5654
http://nrich.maths.org/7277
http://nrich.maths.org/1434
http://nrich.maths.org/292
http://nrich.maths.org/1313
http://nrich.maths.org/1386
Mirrors
How do you hold two mirrors so as to get an integral number of
images of yourself? Discuss the “handedness” of the images.
Set up two mirrors so as to make perfect kaleidoscopic patterns.
How can you use them to make a snowflake?
Fold and cut hearts out of paper. Then make paper dolls. Then
honest snowflakes.
Set up three or more mirrors so as to make perfect kaleidoscopic
patterns. Fold and cut such patterns out of paper.
Why does a mirror reverse right and left rather than up and down?
Find out more
https://www.math.cornell.edu/News/2012-2013/thurston.html
http://www.geom.uiuc.edu/docs/education/institute91/handouts/
handouts.html
On proof and progress in mathematics, Bulletin of the American
Mathematical Society, 30(2), April 1994, pp. 161-177