Ciro Santilli
OurBigBook.com
$£
Sponsor
中国
独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件 邓家贵、低端人口、西藏骚乱
Geometry
Home
Mathematics
Area of mathematics
OurBigBook.com
words: 6k
articles: 167
Table of contents
6k
167
Minimum bounding box
Geometry
Bounding box
Geometry
Fractal
Geometry
Point
(geometry)
Geometry
39
3
Line
(geometry)
Point
Hyperplane
Point
39
1
Plane
(geometry)
Hyperplane
n-sphere
(
S
n
)
Geometry
9
9
Antipodal point
n-sphere
Diameter
n-sphere
1
Radius
Diameter
Circle
(
S
1
, 1-sphere)
n-sphere
6
2
Squaring the circle
Circle
Tarski's circle-squaring problem
(Cut a circle into square)
Circle
6
Sphere
(
S
2
, 2-sphere)
n-sphere
1
Great circle
Sphere
3-sphere
(
S
3
)
n-sphere
3
Projective geometry
Geometry
735
12
Projective space
(
P
(
V
)
)
Projective geometry
58
1
Projective plane
Projective space
Real projective space
(
R
P
n
,
P
(
R
n
+
1
)
)
Projective geometry
677
9
Real projective line
(
R
P
1
,
P
(
R
2
)
)
Real projective space
20
Real projective plane
(
R
P
2
,
P
(
R
3
)
)
Real projective space
640
7
Synthetic geometry of the real projective plane
Real projective plane
245
Model of the real projective plane
Real projective plane
259
2
Lines through origin model of the real projective plane
Model of the real projective plane
5
Spherical cap model of the real projective plane
Model of the real projective plane
254
The real projective plane is not simply connected
Real projective plane
55
Point at infinity
Real projective plane
Homogenous coordinates
Real projective plane
Polytope
Geometry
280
28
Convex polytope
Polytope
Regular polytope
Polytope
257
5
Classification of regular polytopes
Regular polytope
252
4
Simplex
Classification of regular polytopes
35
Hypercube
Classification of regular polytopes
78
1
Hyperrectangle
Hypercube
6
Cross polytope
Classification of regular polytopes
37
Polygon
Polytope
1
12
Quadrilateral
Polygon
1
Rectangle
Quadrilateral
Parallelogram
Polygon
1
2
Parallelepiped
Parallelogram
1
1
Volume of the parallelepiped
Parallelepiped
Regular polygon
Polygon
6
Regular convex polygon
Regular polygon
5
Triangle
Regular convex polygon
Square
Regular convex polygon
Pentagon
Regular convex polygon
Hexagon
Regular convex polygon
Octagon
Regular convex polygon
Polyhedron
Polytope
2
Tetrahedron
Polyhedron
Octahedron
Polyhedron
Regular polyhedron
Polytope
13
1
Platonic solid
Regular polyhedron
13
4-polytope
Polytope
2
Regular 4-polytope
4-polytope
1
Tesseract
Regular 4-polytope
Differential geometry
Geometry
5k
101
Lie group
Differential geometry
5k
100
Lie derivative
Lie group
1
Applications of Lie groups to differential equations
(How to use Lie Groups to solve differential equations)
Lie group
68
Lie algebra
Lie group
738
15
Infinitesimal generator
Lie algebra
147
Lie group-Lie algebra correspondence
Lie algebra
103
7
Lie algebra exponential covering problem
Lie group-Lie algebra correspondence
40
2
A single exponential map is not enough to recover a simple Lie group from its algebra
Lie algebra exponential covering problem
5
The product of a exponential of the compact algebra with that of the non-compact algebra recovers a simple Lie from its algebra
Lie algebra exponential covering problem
22
Two different Lie groups can have the same Lie algebra
Lie group-Lie algebra correspondence
42
3
Every Lie algebra has a unique single corresponding simply connected Lie group
Two different Lie groups can have the same Lie algebra
23
1
Universal covering group
Every Lie algebra has a unique single corresponding simply connected Lie group
6
Every Lie group that has a given Lie algebra is the image of an homomorphism from the universal cover group
Two different Lie groups can have the same Lie algebra
Lie bracket
Lie algebra
Exponential map
Lie algebra
117
1
Exponential map
(Lie theory)
Exponential map
112
Baker-Campbell-Hausdorff formula
(BCH formula)
Lie algebra
153
Generator of a Lie algebra
Lie algebra
Generators of a Lie algebra
Lie algebra
7
Continuous symmetry
Lie group
221
2
Local symmetry
Continuous symmetry
210
1
Local symmetries of the Lagrangian imply conserved currents
Local symmetry
97
Important Lie group
Lie group
3k
67
Matrix Lie group
Important Lie group
284
4
Every closed subgroup of
G
L
(
n
,
C
)
is a Lie group
Matrix Lie group
2
Lie algebra of a matrix Lie group
Matrix Lie group
273
2
Lie bracket of a matrix Lie group
Lie algebra of a matrix Lie group
69
One parameter subgroup
Lie algebra of a matrix Lie group
73
Classical group
Important Lie group
141
3
Symplectic group
(
S
p
(
n
,
F
)
)
Classical group
141
2
Symplectic matrix
Symplectic group
Unitary symplectic group
(
S
p
(
n
)
)
Symplectic group
General linear group
(
G
L
(
n
)
,
G
L
(
n
,
F
)
)
Important Lie group
102
1
Finite general linear group
(
G
L
(
n
,
F
m
)
,
G
L
(
n
,
m
)
)
General linear group
45
Lie algebra of
G
L
(
n
)
Important Lie group
47
Special linear group
(
S
L
(
n
)
)
Important Lie group
534
4
Special linear group of dimension 2
(
S
L
(
2
)
)
Special linear group
Lie algebra of
S
L
(
n
)
Special linear group
486
1
Lie algebra of
S
L
(
2
)
Lie algebra of
S
L
(
n
)
486
Finite special general linear group
(
S
L
(
n
,
m
)
)
Special linear group
37
Isometry group
Important Lie group
102
1
Lie algebra of a isometry group
Isometry group
86
Orthogonal group
(
O
(
n
)
)
Important Lie group
855
23
Definition of the orthogonal group
Orthogonal group
281
5
The orthogonal group is the group of all matrices that preserve the dot product
Definition of the orthogonal group
160
1
What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms
The orthogonal group is the group of all matrices that preserve the dot product
104
The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose
Definition of the orthogonal group
82
1
Elements of the orthogonal group have determinant plus or minus one
The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose
The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns
Definition of the orthogonal group
23
Topology of the orthogonal group
Orthogonal group
257
2
The orthogonal group is compact
Topology of the orthogonal group
Connected components of the orthogonal group
(The orthogonal group has two connected components)
Topology of the orthogonal group
257
Lie algebra of
O
(
n
)
Orthogonal group
3
Special orthogonal group
(
S
O
(
n
)
, Rotation group)
Orthogonal group
206
3
Lie algebra of
S
O
(
3
)
Special orthogonal group
171
1
Lie bracket of the rotation group
Lie algebra of
S
O
(
3
)
16
3D rotation group
(
S
O
(
3
)
)
Special orthogonal group
4
Unitary group
(
U
(
n
)
)
Orthogonal group
108
8
Unitary group of degree 1
(
U
(
1
)
)
Unitary group
Unitary group of degree 2
(
U
(
2
)
)
Unitary group
5
Unit circle
Unitary group
12
Special unitary group
(
S
U
(
n
)
)
Unitary group
44
4
Special unitary of degree 2
(
S
U
(
2
)
)
Special unitary group
9
3
Representations of
S
U
(
2
)
Special unitary of degree 2
4
2
Lie algebra of
S
U
(
2
)
Representations of
S
U
(
2
)
3
2D representation of
S
U
(
2
)
Representations of
S
U
(
2
)
1
Projective linear group
Important Lie group
20
5
Finite projective linear group
(
P
G
L
(
q
,
p
)
)
Projective linear group
Projective special linear group
Projective linear group
8
3
Finite projective special linear group
(
P
S
L
(
p
,
q
)
)
Projective special linear group
8
2
P
S
L
(
2
,
p
)
Finite projective special linear group
8
1
PSL(2,7)
P
S
L
(
2
,
p
)
8
Poincaré group
Important Lie group
1k
17
Galilean transformation
Poincaré group
460
6
Translation
(geometry)
Galilean transformation
139
2
Translation group
Translation
133
1
The derivative is the generator of the translation group
Translation group
121
Galilean invariance
Galilean transformation
321
2
Covariance
Galilean invariance
133
1
Invariant vs covariant
Covariance
43
Lorentz group
(
S
O
(
1
,
3
)
)
Poincaré group
552
9
Representation theory of the Lorentz group
Lorentz group
72
3
Representation of the Lorentz group
Representation theory of the Lorentz group
20
2
Lie algebra of the Lorentz group
Representation of the Lorentz group
Spinor
Representation of the Lorentz group
8
Lorentz boost
Lorentz group
40
Indefinite orthogonal group
(
O
(
m
,
n
)
)
Lorentz group
293
3
Definition of the indefinite orthogonal group
Indefinite orthogonal group
267
1
All indefinite orthogonal groups of matrices of equal metric signature are isomorphic
Definition of the indefinite orthogonal group
135
Indefinite special orthogonal group
(
S
O
(
m
,
n
)
)
Indefinite orthogonal group
15
Representation theory
Lie group
239
3
Irreducible representation
Representation theory
1
Casimir element
Irreducible representation
Schur's lemma
Representation theory
Simple Lie group
Lie group
22
1
Classification of simple Lie groups
Simple Lie group
22
Lie group bibliography
Lie group
135
4
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011)
Lie group bibliography
32
Lie Groups, Physics, and Geometry by Robert Gilmore (2008)
Lie group bibliography
99
Naive Lie theory by John Stillwell (2008)
Lie group bibliography
Lie Algebras In Particle Physics by Howard Georgi (1999)
Lie group bibliography
Tesselation
Geometry
9
5
Aperiodic tiling
Tesselation
Tiling of the plane
Tesselation
9
3
Aperiodic monotile
Tiling of the plane
1
2
Smith aperiodic monotile
(needs reflections, March 2023)
Aperiodic monotile
1
Spectre aperiodic monotile
(no reflections, May 2023)
Aperiodic monotile
Ancestors
(3)
Area of mathematics
Mathematics
Home
Incoming links
(2)
Adolfo Amidei
Positive definite matrix
Ciro Santilli
OurBigBook.com