Geometry

• Space (Euklidian, Hilbert, etc.)

• Dimension (1D, 2D, 3D, 4D, etc.)

A space is a set (universe) with some added structure. While modern math uses many types of spaces (Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, probability spaces, etc), it does not define the notion of "space" itself.

(aside) Undefined concepts in math: number, set, division with zero, point, line, line segment, plane, space, betweeness, incidence, congruence, motion.

Geometric axiomatic system: 1. Primitives (undefined terms) in math, in general, can be objects and relationships. In geometry, the objects are things like points, lines, planes, while a fundamental relation is that of incidence, which is a relation between 2 objects, when one object meets or joins the another. 2. Axioms are statements about these primitives. Axioms are assumed true, and they need not be proven. They are the building blocks of geometric concepts, they specify properties of the primitives. 3. The laws of logic 4. Theorems are the logical consequences of the axioms, the statements that can be derived from the axioms using the laws of deductive logic.

Axiom example: any two points are together incident with one line i.e. for any two points, there is just one line which passes through both of them.

Geometric objects

• point: has coordinates (x, y)

• line: infinite in both directions, defined by 2 points on a plane

• line segment: finite, defined by 2 points on a plane

• ray: infinite in one direction, defined by 2 points on a plane

• angle

• plane (flat plane)

• Polytopes

  • Polygons

  • Polygon is a 2-dimensional Polytope

  • edges + vertices

  • n-gon is a polygon with n sides

area = perimeter/2 * radius

• 2D

  • 3-gon (3 sides):

    • Triangle

      • equilateral triangle

      • right-angled triangle

  • 4-gon (4 sides)

    • Quadrilateral

      • rectangle (stright line sides, equiangular)

      • square (stright line sides, equiangular and equilateral)

      • parallelogram (stright line sides)

      • rhomboid (stright line sides, equiangular)

      • trapezoid

  • Pentagon (a 5-gon)

  • Hexagon (a 6-gon)

  • Oblique closed figure

    • ellipses (r1, r2)

    • circle (r)

    • blob, closed curvature

• 3D:

  pyramid

  cube

  sfere

cuboid parallelepiped ellipsoid

rectilineal plane figure

n-orthotope (hyperrectangle or box) is the generalization of a rectangle for higher dimensions

Regular polygon

In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be either convex or star.

Parallelepiped

In geometry, a parallelepiped is a 3D figure formed by 6 parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square or as a cuboid to a rectangle.

Quadrilaterals

Area

regular polygon half its perimeter times the apothem (apothem is the distance from the center to the nearest point on any side).

Square

• radius = side/2

• perimeter = 4 side = 8 r

• area = perimeter/2 * r = 8r^2

Circle

• r radius

• d diameter = 2r

• circumference = 2π

• area = circumference/2 * r = πr