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Glome in colors | R
What is the shape of the universe? Does it extend infinitely? We can go around the earth forever but can not find the end. Likewise, the universe may be finite and endless. Let's consider a mathematical model suitable for such a space.

Hollow Glome and Luffa | R
A hollow glome can be a counterexample to the Poincaré conjecture. It is simply connected, closed 3-manifold but is not homeomorphic to the 3-sphere or glome.

2018-10-30 | R
2018-10-30 04:25

3-sphere | R
Any loop or circular path on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture provides that the 3-sphere is the only three-dimensional manifold (up to homeomorphism) with these properties.

A Breakthrough in Higher Dimensional Spheres | R
Infinite Series | PBS Digital Studios

Cartesian product | R
For sets A and B, the Cartesian product A × B is the set of all ordered pairs(a, b) where a ∈ A and b ∈ B.

Closed set | R
A set which contains all its limit points. A closed set contains its own boundary.

Compact space | R
Compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded(that is, having all its points lie within some fixed distance of each other).

Connected space | R
A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected.

Contractible space | R
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.

History of the Poincaré Conjecture | R
John Morgan

How Fast Is It - 01 - Preface | R
How Fast Is It - 01 - Preface (1080p)

How Fast Is It - 02 - The Speed of Light | R
How Fast Is It - 02 - The Speed of Light (1080p)

How Fast Is It - 03 - Special Relativity | R
How Fast Is It - 03 - Special Relativity (1080p)

How Fast Is It - 04 - General Relativity I - Geometry | R
How Fast Is It - 04 - General Relativity I - Geometry (1080p)

How the Universe Works | R
Blow your Mind of the Universe Part 11 - Space Discovery Documentary

Images | 11
Figure 7 | R    

Implicit Function | R
2018-12-02 08:49

Implicit function theorem | R
2018-12-02 09:00

Leonard Susskind | R
The space itself may be more than three dimensions. But we can't visualize more dimensions. The architecture of the brain itself is evolved in the world of three dimensions. We only describe more dimensions by pure mathematics.

Loop | R
A loop in mathematics, in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1).

n-ball | R
A set of points whose distance from a particular point is less than a specified length in n-dimensional space.

n-sphere | R
A set of points at a specified distance from a particular point in (n+1)-dimensional space. The pair of points at the ends of a line segment is a 0-sphere.

Non-Euclidean Geometry | R
Classroom Aid

Path | R
In mathematics, a path in a topological space X is a continuous function f from the unit interval I = [0,1] to X f : I → X. A topological space for which there exists a path connecting any two points is said to be path-connected.

Poincaré conjecture | R
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Real coordinate space | R
In mathematics, real coordinate space of n dimensions, written Rn (/ɑːrˈɛn/ ar-EN) (also written ℝn with blackboard bold) is a coordinate space that allows several (n) real variables to be treated as a single variable.

Riemannian Curvature Tensor | R
Classroom Aid

Simply connected space | R
A topological space X is called simply connected if it is path-connected and any loop in X defined by f : S1 → X can be contracted to a point.

Simply connected space | R
A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point, even though it has a "hole" in the hollow center.

Singularities Explained | Infinite Series | R
Singularities Explained | Infinite Series

Space | R
A space consists of selected mathematical objects that are treated as points, and selected relationships between these points.

The Poincare Conjecture | R
In the early 1900s mathematicians and physicists were very interested in the shape of space. New experiments and theories were being developed that would ultimately create relativity theory and change our entire view of the universe.

Thinking visually about higher dimensions | R
Thinking visually about higher dimensions

Three-torus | R
The three-dimensional torus, or triple torus, is defined as the Cartesian product of three circles. The triple torus is a three-dimensional compact manifold with no boundary.

Tom Campbell | R
The assumption that space and time is continuous yields Zeno's paradox. The universe is not created out of continuous space and time.

Topological manifold | R
A topological space which locally resembles real n-dimensional space in a sense defined below.

volumes and surface areas of n-spheres  | R
Graphs of volumes and surface areas of n-spheres of radius 1.

Zeno's Paradox | R
To get to a point in the geometric space, you have to travel half way to that point and half way to that halfway point and so on. There's no end to the halfway so you can't reach the point.

Zeno's Paradox | 4
Defending Zeno's Paradox | R     Stochastic Supertasks | R     Supertasks | R     Zeno's Paradoxes | R    
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