• Chapter 1: Dimension two explains Earth
  Earth
  Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
  
  's coordinate system, and introduces the stereographic projection
  Stereographic projection
  The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
  
  .
• Chapter 2: Dimension three discusses how two-dimensional beings would imagine three-dimensional objects.
• Chapters 3 and 4: The fourth dimension talk about four-dimensional polytope
  Polytope
  In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
  
  s (polychora), projecting the regular ones stereographically on the three-dimensional space.
• Chapters 5 and 6: Complex numbers are about the square root of negative numbers
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
  , transformations, and fractal
  Fractal
  A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
  
  s.
• Chapters 7 and 8: Fibration show what a fibration
  Fibration
  In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
  
   is. Complex numbers are used again, and there are circle
  Circle
  A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
  
  s and tori
  Torus
  In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
  
   rotating and being transformed.
• Chapter 9: Proof emphasizes the importance of proofs
  Mathematical proof
  In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
  
   in mathematics, and proves the circle-conservationess of the stereographic projection as an example.

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