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Book: Geometry

Geometry theorems, Geometries, Geometry illustrations, Universal hyperbolic geometry

几何 _ _ _ _ געאָמעטרי

Overall goals:

I should

Four kinds of geometry

General notions

Path geometry

Line geometry

Angle geometry

Oriented area geometry

Geometry to study

Linear Algebra

Plane Geometry


Affine and Projective Geometry

Affine geometry

Projective geometry

The (n+p-1)-dimensional projective space associated with a quadratic space with signature (n,p), is divided by its (n+p-2)-dimensional surface (images of null vectors), which is a conformal space with signature (n-1,p-1), into 2 curved spaces: one with signature (n-1,p) and positive curvature, the other with dimension (n,p-1) and negative curvature. Just by changing convention, the one with signature (n-1,p) and positive curvature can also seen as a space with signature (p,n-1) and negative curvature; and similarly for the other.

the horizon, or equivalently the polar point of that horizon (the point representing in the projective space the direction orthogonal to that hyperplane). So there are 3 possibilities. The null one sees it as a paraboloid and gives it an affine geometry. The 2 others, with the different signs, see it as a quadric whose center is the polar point, and give it 2 different curved geometries

projective transformation of the space, which changes the sphere into a paraboloid, itself projected into an affine space.

Classical Algebraic Geometry



Modern Algebraic Geometry


Ravi Vakil: The intuition for schemes can be built on the intuition for affine complex varieties. Allen Knutson and Terry Tao have pointed out that this involves three different simultaneous generalizations, which can be interpreted as three large themes in mathematics.







Algebraic Topology




Homology and Cohomology

Geometry and Logic

Differential Geometry

Other Geometry


Noncommutative geometry


Relating Geometries

History of Geometry


Organizing Geometry



Different geometries

Conformal Geometry

Conformal geometry

Hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry

Compare to: Beltrami-Klein model of hyperbolic geometry


Symplectic Geometry


Symplectic geometry

spaces, and measures the sizes of 2-dimensional objects rather than the 1-dimensional lengths and angles that are familiar from Euclidean and Riemannian geometry. It is naturally associated with the field of complex rather than real numbers. However, it is not as rigid as complex geometry: one of its most intriguing aspects is its curious mixture of rigidity (structure) and flabbiness (lack of structure). What is Symplectic Geometry? by Dusa McDuff

mechanical systems, such as a planet orbiting the sun, an oscillating pendulum or a falling apple. The trajectory of such a system is determined if one knows its position and velocity (speed and direction of motion) at any one time. Thus for an object of unit mass moving in a given straight line one needs two pieces of information, the position q and velocity (or more correctly momentum) p:= ̇q. This pair of real numbers (x1,x2) := (p,q) gives a point in the plane R2. In this case the symplectic structure ω is an area form (written dp∧dq) in the plane. Thus it measures the area of each open region S in the plane, where we think of this region as oriented, i.e. we choose a direction in which to traverse its boundary ∂S. This means that the area is signed, i.e. as in Figure 1.1 it can be positive or negative depending on the orientation. By Stokes’ theorem, this is equivalent to measuring the integral of the action pdq round the boundary ∂S.

single point of the plane, but rather lying in a region of the plane. The Bohr-Sommerfeld quantization principle says that the area of this region is quantized, i.e. it has to be an integral multiple of a number called Planck’s constant. Thus one can think of the symplectic area as a measure of the entanglement of position and velocity.


Defining Geometry

Geometry is:

Geometry is the way of fitting a lower dimensional vector space into a higher dimensional vector space.

Definitions of geometry

Geometry challenges

Construction of the continuum

A System of Geometries


Four Basic Geometries

Center and Totality

Generalize this result to n-dimensions (starting with 4-dimensions): Full finite symmetry groups in 3 dimensions


Consider a triangle with 3 directed sides A, B, C:

Four infinite families of polytopes can be distinguished by how they are extended in each new dimension. They seem to relate to four different geometries and four different classical Lie algebras:

Lines, Angles, Areas require a Field whereas directions do not. Lines are translations and Angles are rotations. Together they define the Complexes. Are they key to Dn? Study visual complex analysis. A Field allows, for example, proportionality and other transformations - multiplications - consider!

Projective geometry adds points at infinity to affine geometry. Conformal geometry or inversive geometry adds a distinguished circle. Symplectic geometry adds an area product. Moebius strip plays with the distinguished circle changing orientation if you go around.


Understanding the demicubes

Defining my own demicubes


Distinct Geometries

Special geometries

These geometries show how to relate (ever more tightly) two distinct dimensions.

Pairs of Geometries

Each of the four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space. I imagine that tensors are important as the trivial, "plain vanilla" version of this.

Think of pairs of geometries as defining equivalence classes variously. Equivalence classes are related to actions of symmetry groups.

6 Specifications

The 6 specifications between 4 geometries are transformations which make one geometry more specific than another geometry by introducing orientation, angles and areas. This also makes distance more sophisticated, allowing for negative (oriented) numbers, rational (angular) numbers, and real (continuous) numbers.

Sources to think about

Ideas about transformations





Complex number dilation (rotating).



Other transformations

How do we assign this needed structure? Such a local structure could provide a measure of ‘distance’ between points (in the case of a metric structure), or ‘area’ of a surface (as is speciWed in the case of a symplectic structure, cf. §13.10), or of ‘angle’ between curves (as with the conformal structure of a Riemann surface; see §8.2), etc. In all the examples just referred to, vector-space notions are what are needed to tell us what this local geometry is, the vector space in question being the n-dimensional tangent space Tp of a typical point p of the manifold M (where we may think of Tp as the immediate vicinity of p in M ‘infinitely stretched out’; see Fig. 12.6). Penrose, Road to Reality, page 293, §14.1.

1999. I asked God which questions I should think over so as to understand why good will makes way for good heart. He responded:

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