E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010
Lecture 3: Geometry
Geometry is the science of shape, size and symmetry. While arithmetic dealt with numerical
structures, geometry deals with metric structures. Geometry is one of the oldest mathemati-
cal disciplines and early geometry has relations with arithmetics: we have seen that that the
implementation of a commutative multiplication on the natural numbers is rooted from an inter-
pretation of n×mas an area of a shape that is invariant under rotational symmetry. Number
systems built upon the natural numbers inherit this. Identities like the Pythagorean triples
32+ 42= 52were interpreted geometrically. The right angle is the most ”symmetric” angle
apart from 0. Symmetry manifests itself in quantities which are invariant. Invariants are one
the most central aspects of geometry. Felix Klein’s Erlanger program program uses symmetry
to classify geometries depending on how large the symmetries of the shapes are. In this hour,
we look at a few results which can all be stated in terms of invariants. In the presentation as
well as the worksheet part of this lecture, we will work us through 4 smaller miracles, special
points in triangles as well as 4 gems, the theorems of Pythagoras,Thales,Hippocrates and
Feuerbach. All of these examples illustrate the imp ortance of the concept of symmetry.
Much of geometry is based on our ability to measure length, the distance between two points.
A modern way to measure distance is to determine how long light needs to get from one point
to the other. This geodesic distance generalizes to curved spaces like the sphere and is also a
practical way to measure distances, for example with lasers.It bypasses the problem to determine
first the underlying nature of the space in which we do geometry. Having a distance d(A, B)
between any two points A,B , we can look at the next more complicated object, which is a set
A, B, C of 3 points, a triangle. Given an arbitrary triangle ABC, are there relations b etween the
3 possible distances a=d(B, C ), b =d(A, C), c =d(A, B )? If we fix the scale by c= 1, then
a+b≥1, a + 1 ≥b, b + 1 ≥a. For any pair of (a, b) in this region, there is a triangle. After an
identification, we get the moduli space, an abstract space, which represent all triangles uniquely
up to similarity. We will look at this in the presentation part and a worksheet if time permits.
Asphere is the set of points which have distance 1 from a given point. In the plane, the sphere
is called a circle. A natural problem is to find the circumference L= 2πof a unit circle, the
area A=πof a unit disc, the area F= 4πof a unit sphere and the volume V= 4 = π /3 of a
unit sphere. Measuring the length of segments on the circle leads to new concepts like angle or
curvature. Because the circumference of the unit circle in the plane is L= 2π, angle questions
are tied to π. The most symmetric sit uation of two lines crossing is when all 4 angles which
appear are the same. This leads to the right angle.
Also volumes were among the first quantities, Mathematicians wanted to measure and compute.
For example, a problem on Moscow papyrus dating back to 1850 BC explains the general for-
mula h(a2+ab +b2)/3 for a truncated pyramid with base length a, roof length band height h.
An other great moment of mathematics is the determination of the volume of the sphere by
Archimedes. Place a cone inside a cylinder. The complement of the cone inside the cylinder has
on each height hthe area π−πh2. The half sphere cut at height his a disc of radius (1 −h2)
which has area π(1 −h2) too. Since the slices at each height have the same area, the volume must
be the same. The complement of the cone inside the cylinder has volume π−π/3 = 2π /3, which
is indeed half of the volume of the sphere.
The first geometric playground was planimetry, the geometry in the flat two dimensional space.
Highlights are Pythagoras theorem,Thales theorem,Hypochrates theorem, and Pappus
theorem, which we explore in a worksheet. Discoveries in planimetry are still made today. We
see also a 19’th century 20th century discovery on the work sheet, the Feuerbach theorem. Greek
Mathematics is closely related to history. It starts with Thales goes over Euclid’s era at 300
BC, and ends with the threefold destruction of Alexandria 47 BC by the Romans, 392 by the
Christians and 640 by the Muslims. Geometry was also a place, where the axiomatic method
was brought to mathematics: theorems are proved from a few statements which are called axioms.
The most famous are the 5 axioms of Euclid:
1. Any two distinct points A, B determines a line through Aand B.
2. A line segment [A, B ] can be extended to a straight line containing the segment.
3. A line segment [A, B ] determines a circle containing Band center A.
4. All right angles are congruent.
5. If lines L, M intersect with a third so that inner angles add up to < π, then L, M intersect.
Euclid wondered whether the fifth postulate can be derived from the first 4. He called theo-
rems derived from the first four the ”absolute geometry”. Only much later, with Karl-Friedrich
Gauss and Janos Bolyai and Nicolai Lobachevsky in the 19’th century in Hyperbolic space
the 5’th axiom does not hold. Indeed, geometry can be generalized to non-flat, or even much more
abstract situations. Basic examples are geometry on a sphere leading to spherical geometry or
geometry on the Poincare disc, a hyperbolic space. Both of these geometries are non-Euclidean.
Riemannian geometry, which is essential for general relativity theory generalizes both con-
cepts to a great extent. An example is the geometry on an arbitrary surface. Curvatures of such
spaces can be computed by measuring length alone, which is how long light needs to go from one
point to the next.
An important moment in mathematics was the merge of geometry with algebra. This giant
step is often attributed to Ren´e Descartes. Together with algebra, the subject leads to algebraic
geometry. We will see in this lecture also how algebra allows to automatize proofs.
Here are some examples of geometries which are determined from the amount of symmetry which
is allowed:
Euclidean geometry Properties invariant under a group of rotations and translations
Affine geometry Properties invariant under a group of affine transformations
Projective geometry Properties invariant under a group of projective transformations
Spherical geometry Properties invariant under a group of rotations
Conformal geometry Properties invariant under angle preserving transformations
Hyperbolic geometry Properties invariant under a group of M¨obius transformations
Since time and space on this 2 page summary is up, lets just water our mouths with pictures about
the 4 special points in a triangle and with which we will begin. We want to see first, why in each
of these cases 3 lines intersect in a common point. It is a manifestation of a symmetry present on
the space of all triangles. Some size is constant, if we move on the space of all triangular shapes.
It’s Geometry!
