Glome, Hollow Glome and Luffa
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.
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.
A set which contains all its limit points.
A closed set contains its own boundary.
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).
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.
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.
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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 ...
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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 ...
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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.
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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 ...
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In a glome, the section where z is zero is the projection of a sphere.
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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 ...
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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 ...
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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 ...
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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 ...
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.
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).
A set of points whose distance from a particular point is less than a specified length in n-dimensional space.
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.
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.
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.
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.
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.
A space consists of selected mathematical objects that are treated as points, and selected relationships between these points.
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.
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.
A topological space which locally resembles real n-dimensional space in a sense defined below.