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The eighth proposition in the third book of the Elements is one of Euclid's most complex propositions. It shows that a segment through an outside point D and a circle is ...

An affine geometry is a geometry in which properties are preserved by parallel projection from one plane to another. In an affine geometry, the third and fourth of Euclid's ...

A geometry in which Euclid's fifth postulate holds, sometimes also called parabolic geometry. Two-dimensional Euclidean geometry is called plane geometry, and ...

For any two integers a and b, suppose d|ab. Then if d is relatively prime to a, then d divides b. This results appeared in Euclid's Elements, Book VII, Proposition 30. This ...

An array of "trees" of unit height located at integer-coordinate points in a point lattice. When viewed from a corner along the line y=x in normal perspective, a quadrant of ...

The construction of polyhedra using identical building blocks. The illustrations above show such constructions for the cuboctahedron, octahedron rhombic dodecahedron, and ...

A statement, also known as an axiom, which is taken to be true without proof. Postulates are the basic structure from which lemmas and theorems are derived. The whole of ...

The 21 assumptions which underlie the geometry published in Hilbert's classic text Grundlagen der Geometrie. The eight incidence axioms concern collinearity and intersection ...

In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 2005, p. 3), although this method applies only to even perfect numbers. In a 1638 ...

Given a property P, if P(x)∼x as x->infty (so, using asymptotic notation, the number of numbers less than x not satisfying the property P is o(x), where o(x) is one of the ...