The Logo of the JMC 2008 has a lot of mathematics inside. Here we want to present three different aspects of it.
1. Euclidean Geometry – Intersecting Lines in a Triangle
Consider an arbitrary triangle and its incircle. The three lines segments connecting the points of contact of the incircle with the opposite vertices intersect in a common point.
The proof of this theorem we leave to you as a nice exercise. Note that the segments are not the angular bisections and the common point is not the centre of the incircle.
2. Axiomatic Geometry – The Fano Plane
The idea of axiomatic geometry is to state some axioms and to conclude everything else from these axioms. A geometry in this sense consists of a set points and a set of lines which fulfill such a system of axioms. A common system is that of the projective plane:
A1: Every line contains at least three points.
A2: Every two points lie on a unique line.
A3: Every two lines have a common point.
A4: There exist three non-collinear points (i.e. they do not lie on the same line).
The well known Euclidean geometry violates the third axiom, but it can be transformed into a projective geometry by introducing additional points at infinity. There are models of the projective plane with only a finite number of points and lines. The smallest example is the Fano plane (after Gino Fano). It consists of seven points (red dots) and seven lines (blue segments and the circle).
In the Fano plane on every line lie exactly three points and in every point intersect exactly three lines. Here you can see the principle of dualism, which is a very nice piece of projective geometry. Indeed due to the symmetry in the axioms, you can always exchange 'points' and 'lines' together with 'lie on' and 'intersect'. This yields some interesting connections between dual theorems, for example the theorems of Pascal and Brianchon are projective duals.
3. Hypercomplex Numbers – The Multiplication of Octonions
Complex numbers are introduced as pairs, z=(a,b), of real numbers with the addition rule (a1,b1)+(a2,b2)=(a1+a2,b1+b2) and the multiplication rule (a1,b1)·(a2,b2)=(a1a2−b1b2,a1b2+a2b1). You can write the complex numbers as z=a+bi, where i is called the imaginary unit. With the property i2=−1 both rules become equivalent.
Going one step further, we can build complex numbers of complex numbers, that means structures like z=(a+bi)+(c+di)j, where j is another imaginary unit. With a third imaginary unit k=ij we can write this number as z=a+bi+cj+dk.
Squares of all imaginary units set to −1 , the multiplication of them is cyclic, ij=k, jk=i and ki=j. Multiplying the first equation with i from the left side, we find −j=ik, thus we see that the multiplication is not commutative. These numbers are called quaternions or Hamilton numbers.
Now we can build complex pairs of quaternions and receive numbers with eight components, z=a+bi+cj+dk+eE+fI+gJ+hK, called octonions or Cayley numbers. Their multiplication table is the following:
Using the Fano plane we can draw a convenient mnemonic for remembering the products of all pairs of imaginary units. Multiplication is cyclic on all lines, with a minus sign if ordered against the arrow direction.
Note, that we lose the associativity, for example (ij)E=K, but i(jE)=−K. Only more special variants hold: z(zy)=(zz)y, z(yy)=(zy)y and z(yz)=(zy)z.