Euclidian Geometry

Euclidean geometry is a mathematical system attributed to Euclid, an ancient Greek mathematician, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these.

	Wiki: Euclidean geometry
	Encyclopedia of mathematics: Euclidean geometry

Non-Euclidian Geometry

Non-Euclidean geometry describes spaces where Euclid's famous parallel postulate (that parallel lines never meet) doesn't hold, leading to geometries like Hyperbolic geometry (many parallel lines, triangles < 180 degrees) and Elliptic geometry (no parallel lines, triangles > 180 degrees, like on a sphere).

Non-Euclidean geometry geometries were not widely accepted as legitimate until the 19th century.

Hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry

Spherical geometry or spherics (from Ancient Greek σφαιρικά) is the geometry of the two-dimensional surface of a sphere or the n-dimensional surface of higher dimensional spheres.

	Wiki: Non-Euclidian Geometry
	Wiki: Hyperbolic geometry
	Wiki: Spherical geometry
	Encyclopedia of Mathematics: Spherical geometry
	YT: Visualizing hyperbolic space hyperbolic and spherical geometries
	YT: BBC: Non-Euclidean Geometry [Topics in the History of Mathematics]
	Wiki: Ordered geometry

Quantum Geometry

In quantum gravity, quantum geometry is the set of mathematical concepts that generalize geometry to describe physical phenomena at distance scales comparable to the Planck length.

	Wiki: Quantum Geometry
	MIT News: Physicists measure quantum geometry for the first time (January 13, 2025)
	Nature Physics: Measurements of the quantum geometric tensor in solids (November 25, 2024)
	arXiv: Quantum Geometry in Quantum Materials (a pedagogical review providing an accessible introduction to the concept of quantum geometry)(March 14, 2025)

Noncommutative Geometry

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense.

	Wiki: Noncommutative Geometry
	nLab noncommutative geometry

Mathematical Concepts Essential for understanding Space

	Mathematics Spacial Concepts
	Wiki: Space
	Wiki: Function space

Infinity and Geometry

	Wiki: Infinity
	YT: New Scientist: Mathematicians have just discovered a new infinity, one that doesn’t follow any of the rules of mathematics. It’s so strange, it could change how we understand logic, numbers, and even the universe itself. (December 3, 2025)
	Wiki: Point at infinity
	YT: Example of an interesting Infinite Geometric Series Representation

Incidence Geometry

	Wiki: Incidence Geometry
	Wiki: Incidence structure

Buekenhout Geometry

Buekenhout geometry or diagram geometry is a generalization of projective spaces, Tits buildings, and several other geometric structures

Wiki: Buekenhout Geometry