Topology of our universe, beyond the limits of our senses, mind and technology, glimpse at a higher dimension. We have to live in a world of illusions and filtered information, and our brain does the best it can to form a 3 dimensional reality from a far more complex existence

 

The Poincaré conjecture says that the only three-dimensional surface, or manifold, on which a loop can be shrunk continuously to a point is that of a four-dimensional sphere. To Poincaré this seemed obvious, even though he could not prove it. But most people find it tricky to visualize a sphere in four dimensions, let alone loops on its surface.

Poincaré came upon his conjecture by analogy with the two-dimensional surfaces of three-dimensional shapes. Suppose you were an ant living on the two-dimensional surface of a three-dimensional shape. The topology of your world could be anything in three dimensions - a sphere, a doughnut, even a doughnut with a knot tied in it. But if you trace out a loop in your world and shrink it, you can only be sure it will shrink to a point if you live on the surface of a sphere

Poincaré made the leap from two dimensions to three. Beings, like us, seem to be trapped in the three-dimensional "surfaces" of four-dimensional shapes. And in our universe, it seems that we can shrink any loop we make down to a point. If the universe is finite the Poincaré conjecture would imply that we are apparently living on the surface of a sphere in four dimensions, rather than any other shape.

What would it be like to live on a three-dimensional surface that was not that of a sphere? If you were standing at the centre of a room in this type of world and had the magical ability to move freely through its walls, floor and ceiling, you would encounter similar weird effects experienced by our two-dimensional ant. Suppose you move up through the ceiling, say, you might re-emerge from the floor. In a similar way, walking through the front wall, you might re-emerge from the back wall. This would alert you to the fact that the room is actually a four-dimensional version of a doughnut, called a torus. Unlike the loop we cast on the surface of a sphere, a loop that goes off through the front wall of this room and re-emerges through the back wall could not be shrunk to a point.

In mathematics, a HYPERSPHERE is a sphere having more than three dimensions. Since the early twentieth century, physicists have used this idea of a higher-dimensional sphere to describe a universe in which time is the fourth dimension.

Today, cosmologists say that the universe of relativity and quantum physics can best be understood when seen as a torus, or donut shape. A universe containing black holes, white holes and "wormholes" conforms best to this model.And a torus has the same formula (2pi²r³) as the HYPERSPHERE.
As a model of the universe, the HYPERSPHERE shows how things emerge in time and are enfolded back into fabric of the universe. You can experience this by rotating your HYPERSPHERE through its center.
The HYPERSPHERE also shows how all things in the universe are interconnected, even when they appear to be separate from one another.

 

Where does the HYPERSPHERE occur in nature?
Almost everywhere! The vortex, which is a section of the torus, occurs throughout the natural world-from tornados, whirlpools and electromagnetic fields, the formation of galaxies, the chaos theory, weather phenomena, Internet nodes, Brain cells . And the torus shape is not limited to vortices. An apple, a tree, even a human being all share this same "toroidal" topology.

 

Geometry of the Universe :

general relativity allows for spacetime to be curved, thus the whole Universe may have a non-flat geometry three possible shapes are allowed, flat, positive or negative curvature

Can the Universe be finite in size? If so, what is ``outside'' the Universe? The answer to both these questions involves a discussion of the intrinsic geometry of the Universe.There are basically three possible shapes to the Universe; a flat Universe (Euclidean or zero curvature), a spherical Universe (positive curvature) or a hyperbolic Universe (negative curvature). Note that this curvature is similar to spacetime curvature due to stellar masses except that the entire mass of the Universe determines the curvature. So a high mass Universe can have positive curvature, a low mass Universe might have negative curvature.

 

 

different tests are avalable to determine the curvature of the Universe, such as measuring triangles or parallel lines

All three geometries are classes of what is called Riemannian geometry, based on three possible states for parallel lines   never meeting (flat or Euclidean)   must cross (spherical)   always divergent (hyperbolic)or one can think of triangles where for a flat Universe the angles of a triangle sum to 180 degrees, in a closed Universe the sum must be greater than 180, in an open Universe the sum must be less than 180

 

note that curvature or geometry of the Universe does not determine how it is connected, which is its topology

Standard cosmological observations do not say anything about how those volumes fit together to give the universe its overall shape--its topology. The three plausible cosmic geometries are consistent with many different topologies. For example, relativity would describe both a torus (a doughnutlike shape) and a plane with the same equations, even though the torus is finite and the plane is infinite. Determining the topology requires some physical understanding beyond relativity.

 

a finite Universe, if wrapped, would appear infinite like a box of mirrors

Like a hall of mirrors, the apparently endless universe might be deluding us. The cosmos could, in fact, be finite. The illusion of infinity would come about as light wrapped all the way around space, perhaps more than once--creating multiple images of each galaxy. A mirror box evokes a finite cosmos that looks endless. The box contains only three balls, yet the mirrors that line its walls produce an infinite number of images. Of course, in the real universe there is no boundary from which light can reflect. Instead a multiplicity of images could arise as light rays wrap around the universe over and over again. From the pattern of repeated images, one could deduce the universe's true size and shape.

 

topologies need not be simple, for example a Moebius strip

Topology shows that a flat piece of spacetime can be folded into a torus when the edges touch. In a similar manner, a flat strip of paper can be twisted to form a Moebius Strip.

 

or a Klien bottle

 

The 3D version of a moebius strip is a Klein Bottle, where spacetime is distorted so there is no inside or outside, only one surface.

 

 

deep space observations indicate that the Universe is simply connected

The usual assumption is that the universe is, like a plane, "simply connected," which means there is only one direct path for light to travel from a source to an observer. A simply connected Euclidean or hyperbolic universe would indeed be infinite. But the universe might instead be "multiply connected," like a torus, in which case there are many different such paths. An observer would see multiple images of each galaxy and could easily misinterpret them as distinct galaxies in an endless space, much as a visitor to a mirrored room has the illusion of seeing a huge crowd.

 

however, a large Universe may be connected in complex ways that are not visible to our limited observations

One possible finite geometry is donutspace or more properly known as the Euclidean 2-torus, is a flat square whose opposite sides are connected. Anything crossing one edge reenters from the opposite edge (like a video game see 1 above). Although this surface cannot exist within our three-dimensional space, a distorted version can be built by taping together top and bottom (see 2 above) and scrunching the resulting cylinder into a ring (see 3 above). For observers in the pictured red galaxy, space seems infinite because their line of sight never ends (below). Light from the yellow galaxy can reach them along several different paths, so they see more than one image of it. A Euclidean 3-torus is built from a cube rather than a square.

 

even simple topologies lead to complex connections

 

A finite hyperbolic space is formed by an octagon whose opposite sides are connected, so that anything crossing one edge reenters from the opposite edge (top left). Topologically, the octagonal space is equivalent to a two-holed pretzel (top right). Observers who lived on the surface would see an infinite octagonal grid of galaxies. Such a grid can be drawn only on a hyperbolic manifold--a strange floppy surface where every point has the geometry of a saddle (bottom).

 

 

and all this is connected in 4D spacetime, not simply in 3D space the key to understand the shape of the Universe is its history and dynamics

Its important to remember that the above images are 2D shadows of 4D space, it is impossible to draw the geometry of the Universe on a piece of paper, it can only be described by mathematics. All possible Universes are finite since there is only a finite age and, therefore, a limiting horizon. The geometry may be flat or open, and therefore infinite in possible size (it continues to grow forever), but the amount of mass and time in our Universe is finite.

Measuring Curvature:

determining the global curvature of the Universe, called k, should in principle be easy to determine a positive (k=+1), flat (k=0) and negative (k=-1) Universe make specific predictions for the number of galaxies as a function of distance

 

Measuring the curvature of the Universe is doable because of ability to see great distances with our new technology. On the Earth, it is difficult to see that we live on a sphere. One stands on a tall mountain, but the world still looks flat. One can see a ship come over the horizon, but that was thought to be atmospheric refraction for a long time.Our current technology allows us to see over 80% of the size of the Universe, sufficient to measure curvature. Any method to measure distance and curvature requires a standard `yardstick', some physical characteristic that is identifiable at great distances and does not change with look back time.

 

in practice, the property of lookback time makes curvature measurement a very difficult problem knowledge of some standard yardstick is required, and distance observed makes the timescale involved to be such that galaxies have evolved and changed in size

 

The three primary methods to measure curvature are luminosity, scale length and number. Luminosity requires an observer to find some standard `candle', such as the brightest quasars, and follow them out to high redshifts. Scale length requires that some standard size be used, such as the size of the largest galaxies. Lastly, number counts are used where one counts the number of galaxies in a box as a function of distance.To date all these methods have been inconclusive because the brightest, size and number of galaxies changes with time in a ways that we have not figured out. So far, the measurements are consistent with a flat Universe, which is popular for aesthetic reasons.