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I want to ask a question about the Three-torus model of the universe, as described in wikipedia:

https://en.wikipedia.org/wiki/Three-torus_model_of_the_universe

"is a proposed model describing the shape of the universe as a three-dimensional torus."

I'm not sure i understand the maths of three-dimensional tori, but I want to ask:

Does this model posit that the shape of the universe is that of a ring doughnut? If so, is the universe contained inside the ring doughnut, or is it supposed to exist only on the surface?

Or does the model say that the universe has the shape of a higher dimensional version of a ring doughnut? If so, does it exist inside the higher dimensional ring doughnut, or on its surface?

A three-torus is the boundary of the solid three torus, just like the two-torus is the surface of a solid donut. You can imagine it as a cubical room where each wall/ceiling/floor is a portal to the opposite-facing wall (i.e. the wall to your right is a portal that sends you to your left), but preserves orientation (when you walk out of the portal, your heart is still on your left).

You can also think of it as a world where your position is described by three coordinates (like x, y, and z in Euclidean space), but each coordinate corresponds to an angle on the unit circle. If you go far enough in one coordinate, you loop back to where you started (360 degrees is the same as 0 on a circle).

## Physical Astronomy for the Mechanistic Universe

Aristotelian cosmology was still present in 17th century understanding of the cosmos. This section briefly explores the contributions of Rene Descartes and Isaac Newton to the development of a new mechanical model for describing the relationship between heavenly bodies. In continental Europe, Rene Descartes theory of vorticies served as a powerful conceptual tool for theorizing the nature of the heavens. In England, Isaac Newton developed a universal theory of gravitation that would provide an underlying mechanism for describing a wide range of celestial and terrestrial motions.

## Human Energy Field

The human energy field forms a torus around the central energy channel. The central channel runs from the perineum to the inner most top crown of the head, just in front of the spine. The circumference of the central channel matches the circumference of the circle that is created when the tip of the thumb touches the tip of the index finger on the same hand.

Within the human being, each **chakra**, each acupuncture point, every energy center, is, in itself, a toroidal flow. It flows within itself, in both directions. Each atom, each cell, each organ, each organ system has its own toroidal field and energy flow, and each nests within the other, to create a larger, human torus.

#### Human Energy Field (HEF) – conception about Energy Fields around Human Body. Esoteric name of HEF – Aura.

The human torus connects to larger tori in the same way that the torus of a human cell or molecule connects to the larger human torus. It is part of the torus of the individual’s soul, and of the Earth, and these tori connect to the universal torus. All tori are connected to Source, which is all inclusive and all encompassing.

## A Few of My Favorite Spaces: The Three-Torus

Living in a three-dimensional torus would be a narcissist's dream.

In the Star Trek: The Next Generation episode &ldquoWhere Silence Has Lease,&rdquo the Enterprise flies into a void. Trying to get out, they set a stationary beacon down (never mind what it would mean for the beacon to be stationary in space or how you&rsquod get it to stay there) so they can better measure how far they have gone. As they fly away, the beacon gets farther from them until it starts to get closer. Eventually, they return to where they started.

The Enterprise didn&rsquot do enough exploration of the void for us to know for certain, but it&rsquos quite possible that they had accidentally wandered (or the godlike being Nagilum had pulled them) into a three-torus. Like the two-dimensional torus, which can be represented as a square with opposite sides glued together, the three-torus can be represented as a cube with opposite faces glued together. When you move forward or to the side, you eventually reappear on the opposite face of the cube. When you move up, you eventually reappear at the bottom of the cube.

The two-torus is easy to picture in many different configurations, but the three-torus is harder to visualize. With the two-torus, the square picture is somewhat helpful, but the donut picture gives me a better feel for what it would really be like to live on a torus. For the three-torus, I don&rsquot have access to enough dimensions of visualization to see it holistically like that. I can try by mentally gluing faces together one at a time. First, the top face is glued to the bottom face, making a solid torus with a square cross-section (a square donut, basically). Then we glue the left to the right, which ends up looking like a donut with part of the inside devoured. But the next step is to glue the inside to the outside. This doesn&rsquot work so well in three dimensions.

Often, when a topologist or geometer is asked about applications of her research, she&rsquoll mumble something about the shape of the universe before trying to distract the other person with a pretty picture of the Poincaré disk. (Or is that just me?) But the three-torus may actually have something to do with the shape of the universe. Topology and geometry give us ways to classify all possible three-dimensional shapes, also called three-manifolds. Depending on what properties we can determine that the universe has, we can narrow down the options for the shape of the universe.

I&rsquom not an astrophysicist, and I&rsquom not up to date on the latest measurements that might help us determine the shape of space, so I don&rsquot know what the current thinking on the shape of the universe is. But what if it is a torus? An alternative visualization of the three-torus shows us how weird that would be.

Another visualization of the three-torus. I like to call this one an infinite scaffold. For larger image, click here. Credit: Jeff Weeks.This infinite scaffold distorts one important feature of the three-torus: the three-torus is finite, while this picture looks infinite. But it sheds more light on how disorienting it would be to live in a torus.

Once again, let&rsquos back up a dimension and think about the analogous picture for the two-torus. This would be an infinite grid on a plane rather than just one square. We&rsquod just keep in mind that points that were in the same relative position in two different squares in the plane were actually the same point.

If you lived in a torus (two- or three-dimensional) and looked out from one point, your line of sight might wrap around the torus several times.

A line emanating from a point in the torus, portrayed in three different ways. I drew this in a two-torus instead of a three-torus because my illustration skills only go so far, but you may be able to imagine a similar thing happening in the infinite scaffold picture of the three-torus.If you looked straight up from a point in the three-torus, you&rsquod see the bottom of your own feet. Depending on your angle and visual acuity, your line of sight could theoretically wrap around the torus infinitely many times. If you looked around, you'd see infinitely many copies of yourself. It&rsquos a narcissist&rsquos dream.

If we do live in a three-torus, of course, it is a bit too big for us to see our own rear ends, which is a relief. I wonder, though, if we'll ever know for sure what manifold we inhabit.

The views expressed are those of the author(s) and are not necessarily those of Scientific American.

### ABOUT THE AUTHOR(S)

Evelyn Lamb is a freelance math and science writer based in Salt Lake City, Utah.

## 5 Answers 5

Generally when physicists talk about the universe being finite, they are talking about the existence of an upper bound $R$ on the distance between any two points in space. Such an upper bound could arise in several ways - perhaps the universe has an edge - a boundary which cannot be crossed - or perhaps the universe has the topology of a 3-sphere, and so if one travels sufficiently far in any direction they would eventually return to their starting point. A spatially infinite universe is one which does not have this feature - given any real number $M$ , there exist two points in the universe which are separated by a distance which is greater than $M$ .

A finite universe of course can presumably only host a finite amount of matter. An *infinite* universe could in principle host either a finite or infinite amount of matter. Mainstream cosmology generally assumes the cosmological principle, which states that on sufficiently large scales the distribution of matter in the universe is homogeneous in such cases, an infinite universe would have an infinite amount of matter in it, but of course this principle may not be accurate.

When cosmologists talk about an infinite Friedmann universe, they mean one with infinite spatial volume, finite and uniform matter density (averaged over cosmological scales), and thus infinite matter. There would be galaxies as far as you care to go, even if you could go far beyond what we call the “cosmological horizon”.

Maybe it is best to mention the three major possibilities of "universal shapes" that exist today.

Let me start by saying that cosmologists envision the universe as a four-dimensional spacetime continuum that can exist in a variety of shapes as long as the shape conforms to the field equations of Einstein's general theory of relativity.

Hornet-shaped spacetime.

And let me start with a very strangely shaped universe. The funnel formed spacetime. Look here or here. This is a hornet-shaped (hrnet=curved) universe. That is if we look at two space dimensions because we obviously cannot see a curved three-dimensional space. This universe allegedly explains observations made on the universe. This is a finite universe. Here's a picture (note that the points of space where the ship turns back inward are problematic):

A spherical spacetime.

The most simple and most common form. This is the "balloon universe", due to the fact that the two-dimensional space part can be viewed as a balloon shape, which can be viewed to represent a curved form of a two-dimensional space.

This universe can be finite or infinite in spatial extent. And these spacetimes can have zero, negative or positive curvature.

Zero curvature corresponds to a spacetime that will continue to expand forever and this expansion will stop "after" infinite time, while its size is infinite. The balloon will stop expanding at infinity and have an infinite size.

The positive curvature represents a universe that expands and after a finite time, this expansion will stop. This universe obviously has a finite size. The balloon grows in size and will contract after some time (spacetime generally cannot stay motionless).

The negative curvature represents a spacetime that will expand at an ever-increasing rate. The ballon starts expanding and will grow in size faster and faster. It looks as if our own universe is (in these days) growing with this last feature.

In our universe, inflation has been assigned to a newly formed balloon spacetime. Initially, the universe expanded at an incredible rate and basically, the whole universe was formed during inflation. Since the end of inflation, the universe has only grown "only" three times as large as in the beginning. So what we see today is just a very tiny fraction of the whole. Behind the horizon the is a vast space.

The three universes are also referred to as flat, closed, or open:

The torus-shaped universe.

In 1984 this model was proposed as the [three-torus model]. 5This universe has a two-dimensional space equivalent of a torus. It has positive as well as negative curvature and can be finite or infinite (the infinite size will be reached after infinite time).

One can invent more shapes, of course, but they have to pass the tests of observation. I'm not aware that another form as the three mentioned is present though.

It is generally accepted (i.e., by most cosmologists) that our universe has a balloon=like topology. The amount of matter in these universes can be finite or infinite. In closed universes, the amount of matter will be finite while this amount is infinite in (asymptotically) flat or open universes.

What about time? Is spacetime eternal? That depends of course if there was something before the big bang (when the spacetimes started evolving very small size). For example, according to Smolin, there was a contracting universe present before the big bang (likewise he proposes that upon entering a black hole there will follow an exit to another spacetime, where there is a white hole present which ejects all that has fallen in from our own spacetime). It is generally claimed though that spacetime had a beginning in time and dependent on how much matter is in it and on the expansion speed, the universe will be there forever or not. It is thought (based on observations) that our universe is nearly flat and will expand forever.

So, what does the concept of an infinite universe mean? In the context of spacetimes that might be clear now: a universe is infinite if the spacetime has an infinite extent. Infinite space goes hand in hand with infinite time. The amount of matter contained in these universes can be finite or infinite.

## Astrophysics team lights the way for more accurate model of the universe

Light from distant galaxies reveals important information about the nature of the universe and allows scientists to develop high-precision models of the history, evolution and structure of the cosmos.

The gravity associated with massive pockets of dark matter that lie between Earth and these galaxies, however, plays havoc with those galactic light signals. Gravity distorts galaxies' light -- a process called gravitational lensing -- and also slightly aligns the galaxies physically, resulting in additional gravitational lensing light signals that contaminate the true data.

In a study first published Aug. 5 in *The Astrophysical Journal Letters*, University of Texas at Dallas scientists demonstrated the first use of a method called self-calibration to remove contamination from gravitational lensing signals. The results should lead to more accurate cosmological models of the universe, said Dr. Mustapha Ishak-Boushaki, professor of physics in the School of Natural Sciences and Mathematics and the corresponding author of the study.

"The self-calibration method is something others proposed about 10 years ago many thought it was just a theoretical method and moved away from it," Ishak-Boushaki said. "But I intuitively felt the promise. After eight years of persistent investigation maturing the method itself, and then the last two years applying it to the data, it bore fruit with important consequences for cosmological studies."

**A Lens on the Universe**

Gravitational lensing is one of the most promising methods in cosmology to provide information on the parameters that underlie the current model of the universe.

"It can help us map the distribution of dark matter and discover information about the structure of the universe. But the measurement of such cosmological parameters can be off by as much as 30% if we do not extract the contamination in the gravitational lensing signal," Ishak-Boushaki said.

Due to the way distant galaxies form and the environment they form in, they are slightly physically aligned with the dark matter close to them. This intrinsic alignment generates additional spurious lensing signals, or a bias, which contaminate the data from the galaxies and thus skew the measurement of key cosmological parameters, including those that describe the amount of dark matter and dark energy in the universe and how fast galaxies move away from each other.

To complicate matters further, there are two types of intrinsic alignment that require different methods of mitigation. In their study, the research team used the self-calibration method to extract the nuisance signals from a type of alignment called intrinsic shape-gravitational shear, which is the most critical component.

"Our work significantly increases the chances of success to measure the properties of dark energy in an accurate way, which will allow us to understand what is causing cosmic acceleration," Ishak-Boushaki said. "Another impact will be to determine accurately whether Einstein's general theory of relativity holds at very large scales in the universe. These are very important questions."

**Impact on Cosmology**

Several large scientific surveys aimed at better understanding the universe are in the works, and they will gather gravitational lensing data. These include the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST), the European Space Agency's Euclid mission and NASA's Nancy Grace Roman Space Telescope.

"The big winner here will be these upcoming surveys of gravitational lensing. We will really be able to get the full potential from them to understand our universe," said Ishak-Boushaki, who is a member and a convener of the LSST's Dark Energy Science Collaboration.

The self-calibration method to remove contaminated signals was first proposed by Dr. Pengjie Zhang, a professor of astronomy at Shanghai Jiao Tong University and a co-author of the current study.

Ishak-Boushaki further developed the method and introduced it to the realm of cosmological observations, along with one of his former students, Michael Troxel MS'11, PhD'14, now an assistant professor of physics at Duke University. Since 2012 the research has been supported by two grants to Ishak-Boushaki from the National Science Foundation (NSF).

"Not everyone was sure that self-calibration would lead to such an important result. Some colleagues were encouraging some were skeptical," Ishak-Boushaki said. "I've learned that it pays not to give up. My intuition was that if it was done right, it would work, and I'm grateful to the NSF for seeing the promise of this work."

## The Models of the Universe: Eudoxus, Aristotle, Aristarchus, Ptolemy, and Copernicus

Eudoxus of Cnidus (born c. 395 – 390 B.C.), a Greek astronomer and mathematician, was the first to propose a model of the universe based on geometry. His model composed of 27 concentric spheres with Earth as the center. The Sun, the Moon, the planets, and the fixed stars have spheres. Each sphere is attached to a larger sphere through a pole. The rotation of the spheres on their poles once every 24 hours accounts for the daily rotation of the heavens. It is unclear whether Eudoxus regarded these spheres as physical entities or just mathematical constructions.

### Aristotle’s Model

Aristotle (born c. 384 B.C.), a Greek philosopher and astronomer, considered the model proposed by Eudoxus, but he considered these spheres as physical entities. He thought that these spheres were filled with the divine and eternal “ether” that caused the spheres to move. He introduced the **Prime Mover**, as the cause of the movement of the spheres. His model composed of 56 spheres that guided the motion of the Sun, the Moon, and five known planets. As the spheres move, they maintained the same distance from the Earth. Also, they moved at constant speeds.

### Aristarchus’ Model

Aristarchus of Samos (born c. 310 B.C.), a Greek astronomer and mathematician, was the first to hypothesize that the Sun is the center of the universe. He visualized that the Moon orbits around a spherical Earth which then revolves around the Sun. He believed that the stars are very far away from the Earth as evidenced by the absence of stellar parallax – that is, the stars do not change positions relative to each other as the Earth revolves around the Sun.

Through geometrical models and mathematical computations, he concluded that the Sun is 20 times farther from the Earth than the Moon is to the Earth the Earth is about three times larger than the Moon and the Sun is 20 times larger than the Moon. He also reasoned out that smaller spheres orbit around larger ones. Thus, the Moon orbits around the Earth, and the Earth orbits around the Sun.

**Ptolemy’s Model**

The Sun, Moon, stars, and planets were believed to move in a uniform circular motion – the “perfect” motion assigned to celestial bodies by the ancient Greeks. However, observations showed otherwise. The paths of the celestial bodies are not circular, and they vary in distances. Babylonians even showed that some planets exhibit a retrograde motion – a motion opposite to that of other planets.

To explain “imperfect motions” of heavenly bodies, **Claudius Ptolemy** (born c. 90 A.D.), a Greco-Egyptian astronomer and mathematician, proposed his own geocentric (Earth-centered) model of the universe. He accounted for the apparent motions of the planets around the Earth by assuming that each planet moved around a sphere called an **epicycle**. The center of the epicycle then moved on a larger sphere called a **deferent**.

**The Ptolemaic System**

- A planet moves counter-clockwise around the epicycle.
- The epicycle’s center also moves counter-clockwise around the center of the deferent (indicated by the + sign in the image).
- The center of the epicycle moves around the equant with a uniform speed.
- The Earth is not exactly at the center of the deferent, or it is eccentric (off the center). This explains why, as observed from the Earth, the Sun or a planet moves slowest when it is farthest from the Earth and moves fastest when it is nearest the Earth.
- The motion of the planet can be described by points 1-7 in the figure below. At point 4, the planet moves in a retrograde (clockwise) motion. The planet is brightest at this point because it is closest to the Earth.

### Copernicus’ Model

In 1543, Nicolaus Copernicus, a Renaissance mathematician and astronomer born in Poland, ended the geocentric astronomy era by publishing his work *On the Revolutions of the Heavenly Spheres* wherein he explained that the Sun, not the Earth, is the center of the universe.

In his work, he reiterated the ancient Greek concept that the motion of spherical heavenly bodies is uniform, eternal, and circular. He then reasoned that because Earth is spherical, then its motion is circular. He added that the Earth has three different motions: daily rotation on its axis, yearly motion around the Sun, and the precession, or change in orientation, of its axis every 26 000 years.

He also proposed that the fixed stars are immovable. Their apparent movement is a consequence of the Earth’s rotation. These stars are at immeasurable distances from the Earth, so there is no observable parallax.

By placing the Sun at the center of the universe and the orbits of Mercury and Venus in between the Sun and the Earth, Copernicus’ model was able to account for the changes in the appearances of these planets and their retrograde motions. The need for epicycles in explaining motions was eliminated.

## Quantum graviton creation in a model universe ☆,☆☆

Quantization in the spatially inhomogeneous, anisotropic, Gowdy three-torus solution of Einstein's equations leads to the production of gravitons from empty space. The creation of pairs of gravitons occurs only in wave modes with wavelength exceeding the horizon size at an initial time. The final number of created gravitons in any mode is proportional to the number of causally unconnected regions at the initial time over the wavelength of that mode. At large times, graviton number is well defined since the solution is in WKB form. The creation process produces the anisotropic collisionless radiation identical to that discussed by Doroshkevich, Zelɽovich, and Novikov which characterizes the large time classical solution. Near the singularity, the model behaves like an empty Bianchi Type I universe at each point in space (local Kasner). The canonical methods of Arnowitt, Deser, and Misner yield a reduced Hamiltonian from which the classical equations of motion are obtained. The quantization of the rapidly varying gravitational field component resembles the procedures used by Parker or Zelɽovich *et al.* to study particle creation in curved spacetime.

Work supported in part by NASA Grant No. NGR-21-002-010.

Portions of this paper form part of Ph.D. thesis submitted to the University of Maryland.

Present address: Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado 80302.