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Glome, Hollow Glome and Luffa

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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

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

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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

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

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Closed set

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

5253#3476 SIBLINGS CHILDREN 3476

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

5253#3476 SIBLINGS CHILDREN 3476

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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

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

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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

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

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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

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

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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

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

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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

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

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Glome in colors 04

All paths are reflected at the boundary of the glome making the space bounded without a boundary. Every path passing through any point in the glome is continuous at the point even at the boundary. This is analogous to the projection of a sphere.

5253#9045 SIBLINGS CHILDREN COMMENT 9045

All paths are reflected at the boundary of the glome making the space bounded without a boundary. Every path passing through any point in the glome is continuous at the point even at the boundary. This is analogous to the projection of a sphere.

5253#9045 SIBLINGS CHILDREN COMMENT 9045

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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

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

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Glome in colors 09

In a glome, the section where z is zero is the projection of a sphere.

5253#9051 SIBLINGS CHILDREN COMMENT 9051

In a glome, the section where z is zero is the projection of a sphere.

5253#9051 SIBLINGS CHILDREN COMMENT 9051

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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

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

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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

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

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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

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

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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

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

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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

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

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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

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

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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

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

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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

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

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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

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

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Poincaré conjecture

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

5253#3579 SIBLINGS CHILDREN 3579

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

5253#3579 SIBLINGS CHILDREN 3579

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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

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

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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

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

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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

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

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Space

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

5253#5251 SIBLINGS CHILDREN 5251

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

5253#5251 SIBLINGS CHILDREN 5251

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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

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

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Thinking visually about higher dimensions

Thinking visually about higher dimensions

5253#3388 SIBLINGS CHILDREN 3388

Thinking visually about higher dimensions

5253#3388 SIBLINGS CHILDREN 3388

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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

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

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Topological manifold

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

5253#3493 SIBLINGS CHILDREN 3493

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

5253#3493 SIBLINGS CHILDREN 3493

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volumes and surface areas of n-spheres

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

5253#3448 SIBLINGS CHILDREN 3448

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

5253#3448 SIBLINGS CHILDREN 3448