## §ummary | ## Summary of Results |

A **Geometry** consists of a set *S* and a subgroup *G* of the group *Bij*(*S*) of all bijections from *S* to itself.

For Euclidean geometry, *S* = **R**^{n} and the group is the set *I*(**R**^{n}) of all length preserving maps or **isometries**.

Every element of *I*(**R**^{n}) is of the form *T*_{a} *L* with *T*_{a} a translation **x** **a** + **x** and *L* a length preserving linear map ∈ *O*(*n*).

Then *det*(*L*) = ±1. If *det*(*L*) = +1 the isometry is **direct**, otherwise it is **opposite**.

If *f* is a symmetry of the line: *f* ∈ *I*(**R**), then *f*(*x*) = *a* + *x* or *f*(*x*) = *a* - *x* and *f* is either a translation or reflection (in a point).

*Direct* isometries of the plane: *I*(**R**^{2}) are either *rotations* (about a point) or *translations*.

*Opposite* symmetries of **R**^{2} are either *reflections* (in a line) or *glides* (a reflection in a line followed by translation parallel to the line).

*Direct* isometries of **R**^{3} are either *rotations* (about an *axis* in **R**^{3}) or *translations* or *screws* (a rotation about a line followed by translation parallel to the line).

*Opposite* symmetries of **R**^{3} are either *reflections* (in a plane) or *glides* (a reflection in a plane followed by translation parallel to the plane) or *rotatory reflections* (rotaion about a line followed by reflection in a plane perpendicular to the line).

A symmetry group of a figure *F* **R**^{n} is the set of all symmetries *f* in *I*(**R**^{n}) for which *f*(*F*) = *F*.

Finite subgroups of *I*(**R**^{2}) are isomorphic either to *C*_{n} : a cyclic group of order *n* (generated by a rotation by 2π/*n* about some point) or *D*_{n} : a dihedral group of order 2*n* (generated either by a similar rotation and one reflection in a line through the point, or by two such reflections) and this classifies the subgroups up to conjugacy inside the group *I*(**R**^{2}).

Finite subgroups of *I*(**R**^{3}) which consist only of direct symmetries are isomorphic either to *C*_{n} (generated by a rotation by 2π/*n* about some axis) or *D*_{n} (a dihedral group of order 2*n* consisting of rotations as in the *C*_{n} case and rotations by π) or to the rotational symmetry groups of one of the Platonic solids (*A*_{4} , *S*_{4} or *A*_{5}).

A finite subgroup of *I*(**R**^{3}) which contains opposite symmetries is either a *direct product* of a direct symmetry group with the group of order 2 generated by the central inversion map *J*: **x** -**x** or a *mixed group*. A **mixed group** is denoted by a pair *H* *K* where *H* and *K* are direct symmetry groups and *K* is a subgroup of index 2 in *H*. The direct symmetries in the mixed group are the elements of *H* and the opposite symmetries are the composition of *J* with the elements of *K* - *H*.

**Affine geometry** is the geometry with group *A*(**R**^{n}) acting on the set **R**^{n}. The elements of this consist of compositions of translations with elements of *GL*(*n*, **R**). Affine transformations do not preserve length or angle, but do preserve ratios of segments on parallel lines.

Many theorems of Euclidean geometry are in fact affine theorems. These include results like the concidence of the medians of a triangle, Ceva's theorem and Menelaus's theorem.

**Similarity geometry** lies between affine and Euclidean geometry. Its transformations are *similitudes* which consist of compositions of translations with linear maps of the form *λ**L* with *L* ∈ *O*(*n*) and *λ* ∈ **R** - {0}. Such transformations preserve angles but stretch all lengths by the same amount.

Theorems involving similar triangles are theorems of similarity geometry.

The **Real projective line** **R***P*^{1} consists of the affine line **R** together with an extra *point at infinity*.

The **Real projective plane** **R***P*^{2} consists of the affine plane **R**^{2} together with a *line at infinity*.

The projective line is parametrised by pairs [*x*, *y*] of **homogeneous coordinates**. An ordinary (affine) point *x* has homogeneous cordinates [*x*, 1] and the point at infinity has homogeneous coordinates [1, 0].

The *projective transformations* acting on **R***P*^{1} are elements of the group *PGL*(2, **R**) which is the quotient group of the group *GL*(2, **R**) by the subgroup of scalar matrices *λ**I*.

These transformations can be regarded as *rational maps* of the form *x* (*ax* + *b*)/(*cx* + *d*) with appropriate allowance for the point at infinity.

The **standard reference points** of are ∞, 0, 1 ([1, 0], [0, 1], [1, 1]) and there is a unique projective transformation taking any three distinct points to these reference points.

Given four points *a*, *b*, *c*, *d* in **R***P*^{1} let *θ* be the map taking *a*, *b*, *c* to ∞, 0, 1. Then the **cross-ratio** (*a*, *b* ; *c*, *d*) is defined to be *θ* (*d*). This cross-ratio is preserved by projective transformations.

The projective plane **R***P*^{2} is parametrised by homogeneous coordinates [*x*, *y*, *z*]. Its group of transformations is *PGL*(3, **R**).

These transformations preserve the cross-ratio of any four points on a line. Any two such *ranges* on different lines which are related by projection from a point not on either line have the same cross-ratio. One may also define the cross-ratio of a *pencil* of four lines (four lines through a common point).

One may use the freedom to project some points to (say) infinity to simplify diagrams and deduce projective theorems from simple Euclidean results. Some examples are the harmonic property of a complete quadrangle, Desargues' theorem, Pappus's theorem.

**Topology** is the study of properties invariant under continuous deformation.

Spaces are **homeomorphic** if there is a continuous bijection with a continuous inverse between them.

A **surface** is a space in which every point lies in an area homeomorphic to a disc in **R**^{2}.

Examples of surfaces include spheres, tori, Klein bottles and Real projective planes. Surfaces like cylinders or Möbius bands which are not closed are examples of *non-compact* surfaces.

More complicated surfaces can be made using the **connected sum construction** in which discs are removed from two surfaces and the pieces then joined by the free edges.

A compact surface can be constructed from a **planar model**: a polygon whose sides are matched in pairs. The matching can be described using an *edge word* which describes how the matched edges are located on the boundary.

If every edge in the edge word is matched by its inverse, the surface is orientable. The edge word of a connected sum is constucted by concatenating the edges words of the individual surfaces.

The **Euler characteristic** *χ* of a surface is calculated by splitting the surface up into *F* regions (or faces), *E* edges and *V* vertices where *χ* = *F* - *E* + *V*. It is a topological invariant of the surface.

The **Classification Theorem** for compact connected surfaces states that if such a surface is orientable it is either a sphere or a connected sum of Tori: if non-orientable it is a connected sum of Projective planes.

We can use it to identify a surface from the edge word of a planar model by calculating the number of vertices on the boundary of the planar model.