• 2018.10.30
2018-10-30 04:25
5253#3449 SIBLINGS CHILDREN 3449

• 3-sphere
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.
5253#3447 SIBLINGS CHILDREN 3447

• Cartesian product
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. 
5253#3619 SIBLINGS CHILDREN 3619

• Closed set
A set which contains all its limit points. A closed set contains its own boundary.
5253#3476 SIBLINGS CHILDREN 3476

• Compact space
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). 
5253#3494 SIBLINGS CHILDREN 3494

• Connected space
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.
5253#3492 SIBLINGS CHILDREN 3492

• Contractible space
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.
5253#3578 SIBLINGS CHILDREN 3578

• Glome in colors 01
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 but endless. Let's consider a mathematical model suitable for such a space. A summary of Eucli ...
5253#707 SIBLINGS CHILDREN COMMENT 707

• Glome in colors 03
In a one-dimensional space, there is only one direction. So it is straight. If a one-dimensional space is looped, the relationship between points can be represented by a closed curve. But the curve shows just the relationship between points, not the shape ...
5253#9044 SIBLINGS CHILDREN COMMENT 9044

• Glome in colors 06
In the projection of a sphere, each point on the disc except the circumference represents two points. One is a point on the upper hemisphere and the other is a point on the lower hemisphere. Likewise, each point in the glome except the surface represents ...
5253#9048 SIBLINGS CHILDREN COMMENT 9048

• Glome in colors 10
Going straight is represented by following an ellipse on the great section and the movement from one hemiglome to another is represented by the contact between the ellipse and the circumference of the great section. For every direction at any point in a g ...
5253#9052 SIBLINGS CHILDREN COMMENT 9052

• Glome in colors 15
A circular model has two dimensions, but the space represented by the model has only one dimension. The spheric model has three dimensions, but there are only two dimensions in the space that the model represents. Likewise, the space represented by the gl ...
5253#9058 SIBLINGS CHILDREN COMMENT 9058

• Glome in colors 18
For any two points in a glome, there is a plane containing the two points and the origin. By cutting the glome with the plane we get a great section where we can draw a great ellipse passing through the two points. This means that at any point in a glome ...
5253#9061 SIBLINGS CHILDREN COMMENT 9061

• History of the Poincaré Conjecture
John Morgan
5253#5225 SIBLINGS CHILDREN 5225

• Hollow Glome and Luffa 06
Suppose a three-dimensional space which is bounded and path-connected. Let the space contain holes. Let f and g be continuous functions taking this space as a domain. f(x,y,z) is equal to g(x,y,z) at the boundary of the space and is different at the other ...
5253#9069 SIBLINGS CHILDREN COMMENT 9069

• How Fast Is It - 01 - Preface
5253#5360 SIBLINGS CHILDREN 5360

• How Fast Is It - 02 - The Speed of Light
5253#5359 SIBLINGS CHILDREN 5359

• How Fast Is It - 03 - Special Relativity
5253#5358 SIBLINGS CHILDREN 5358

• How Fast Is It - 04 - General Relativity I - Geometry
5253#5355 SIBLINGS CHILDREN 5355

• Implicit Function
2018-12-02 08:49
5253#3572 SIBLINGS CHILDREN 3572

• Implicit function theorem
2018-12-02 09:00
5253#3651 SIBLINGS CHILDREN 3651

• Leonard Susskind
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.
5253#5242 SIBLINGS CHILDREN 5242

Pillow English Listening

◌◌◌ Loop
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).
5253#3450 SIBLINGS CHILDREN 3450

• n-ball
A set of points whose distance from a particular point is less than a specified length in n-dimensional space.
5253#3445 SIBLINGS CHILDREN 3445

• n-sphere
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.
5253#3446 SIBLINGS CHILDREN 3446

• Non-Euclidean Geometry
Classroom Aid
5253#3389 SIBLINGS CHILDREN 3389

◌◌◌ Path
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. 
5253#3453 SIBLINGS CHILDREN 3453

• Poincaré conjecture
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
5253#3579 SIBLINGS CHILDREN 3579

• Real coordinate space
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. 
5253#3495 SIBLINGS CHILDREN 3495

• Riemannian Curvature Tensor
Classroom Aid
5253#3394 SIBLINGS CHILDREN 3394

• Simply connected space
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.
5253#3534 SIBLINGS CHILDREN 3534

• Simply connected space
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.
5253#3577 SIBLINGS CHILDREN 3577

• Singularities Explained | Infinite Series
5253#3563 SIBLINGS CHILDREN 3563

◌◌◌ Space
A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. 
5253#5251 SIBLINGS CHILDREN 5251

• The Poincare Conjecture
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.
5253#3533 SIBLINGS CHILDREN 3533

• Thinking visually about higher dimensions
Thinking visually about higher dimensions
5253#3388 SIBLINGS CHILDREN 3388

• Three-torus
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.
5253#3612 SIBLINGS CHILDREN 3612

• Topological manifold
A topological space which locally resembles real n-dimensional space in a sense defined below.
5253#3493 SIBLINGS CHILDREN 3493

• volumes and surface areas of n-spheres 
Graphs of volumes and surface areas of n-spheres of radius 1.
5253#3448 SIBLINGS CHILDREN 3448