Our senses tell us that the world around us, the physical world, consists of objects embedded in space and that events happen at measurable times. A previous post, "On the Nature of Physical Reality—A Mental Construct", discusses some of the philosophy, physical, and spatial aspects of physical reality. There is much more to reality than this aspect of physical reality. Here discussed are the mathematical aspects of spaces, which are characterized by internal relationships that amount to geometries. It turns out that the idea of geometry not only applies to abstract figures drawn on flat pieces of paper, but to the structure of space and time itself and to physical reality even beyond that.

The ancient Egyptians, in the process of carrying out the complex processes of civilization, such as distributing land and constructing buildings, found it convenient to plan their efforts on flat surfaces such as sheets of papyrus. In the process, they discovered a few simple rules governing relationships between aspects of various figures of interest that could be drawn on a flat surface. [See the post, "Geometry and Earth Measure".] Alexander the Great ' s conquest of Egypt brought ancient Greeks in contact with Egyptian thought. The logic of the Greeks and the knowledge of the Egyptians cross fertilised each other resulting in establishing Egypt as a great center of learning in the Greek world. The tangible result of this was the establishment of the famous, ancient library of Alexandria.

From Egyptian rules of thumb, **Euclid of Alexandria** (a Greek), drew out abstract knowledge of spatial relations, called geometry (now known as plain geometry), putting together all that was then known about geometry in a book called "The Elements", which became the theoretical basis for the mathematics of figures drawn on a flat surface. For centuries, this work served as the text book for geometry.

In more modern times a French mathematician, Rene Descartes (1596 –1650) was able to link Euclid's geometry to the rest of mathematics through a branch of mathematics he developed, called analytic geometry, the key idea of which is to index each point in space by a unique set of coordinates that refer to a frame of reference. A set of two numbers is required to index each point in a two dimensional space; three numbers, in a three dimensional space; and n numbers, in an n-dimensional space. Because it takes only two coordinates to specify a point on a plane, Euclid' s geometry was a geometry of two dimensional spaces, but not of all possible two dimensional spaces as is pointed out below. Mathematicians following him, later generalized Euclid's ideas to include three dimensional spaces in a field of study called solid geometry. Nowadays, mathematicians have realized that Euclid' s geometry for two dimensions can be extend to any number of dimensions (n-dimensional spaces). Although figures in the n-dimensional geometry can no longer be drawn on flat surfaces, mathematicians have been able to define the flatness in a way that can be applied to n-dimensional spaces.

The curvature of a space can be diagnosed by geometrical relationships within that space. Curvature effects are more complicated than the effects of bending that space, which could leave a flat surface still "flat". For example, rolling up a flat piece of paper containing all kinds of geometrical figures still leaves those figures invariant within that two dimensional surface.

The concept of a "flat space" implies that there are also "curved spaces". This is a lesson that was learned by humanity by the hard knocks of experience. [See the post, "Geometry and Earth Measure".] Even so, flat spaces can be quite abstruse. For example, consider Einstein's formulation of the Special Theory of Relativity, which is applied to a flat space representation of space and time that has negative portions of its metric (Riemannian geometry). For reference, see a previous article posted in this blog, "Special Relativity II—standard notation."

Although the idea of of applying Riemannian geometry to physical reality was already *way out there*, it was a neat trick that allowed Einstein to think still much farther out of the box. Riemannian geometry was essentially "flat." By allowing that geometry to have intrinsic curvature, Einstein was able to automatically include the effects of gravity as geometrical effects of the warping of the space-time manifold.

Rene Descartes's generalization of Euclid' s geometry through coordinate systems resulted in a branch of mathematics called analytic geometry. Gaspard Monge ,included methods of calculus to further develop another powerful area of mathematics called differential geometry. Methods of differential geometry allowed people to determine the nature of a manifold from relationships determined entirely within its surface. The ideas of Monge were further elaborated by the famous, great mathematician Carl Friedrich Gauss.

In more modern times, Einstein generalized Descartes' s idea to the description of an event as a point in a four dimensional space that is specified by four coordinates: three for space and one for time. The set of all possible events constitute a four dimensional manifold, or space, sometimes referred to as space-time, which conforms to a four dimensional geometry. The theory of Special Relativity deals with the geometrical effects of dynamics in a four dimensional space. Einstein's General Theory of Relativity departs from the flat manifold version of Relativity to allow curvature effects of that manifold. [See earlier post, Special Relativity II—standard notation.]

What we observe on a microscopic scale are events that are the basic coinage of all physical transactions. An event is characterized by where and when something happens.

Classical physics describes an event as a point in three dimensional space, which is specified by three coordinates plus another number that specifies the time of its happening. Therefore individual events are described classically in terms of four coordinates. A four dimensional space-time point describes all possible events--in total a four dimensional space-time continuum. Classically, a space containing all potential event points satisfies a geometry of four dimensional space-time, which was Einstein's starting point for his General Theory of Relativity. In this theory, gravity is treated as a field of distortions of space-time away from a flat manifold as a its curvature, somewhat like the scale factors of map projections. We ordinary humans cannot see this four dimensional manifold globally as a surface embedded in a higher dimensional space because our observations are made within it; however, using methods of differential geometry allows us to analyse such spaces and to extract curvature effects from that analysis.

Currently, some physicists are trying to develop a comprehensive theory of physics that uses strings instead of points for specifying events of elementary object of physics. Perhaps this is a mathematical way of allowing events in out four dimensional manifold to connect with higher dimensions not available to direct observation. There is a lot of interesting philosophy here.

String Theories incorporate more than four dimensions in describing the dynamics of fundamental 'particles', which may be 26 dimensions, or 11 dimensions in M-theory. Reading between the lines, we see that this theory suggests that we do not experience the full dimensionality of physical reality because all but four dimensions of the strings are rolled up into very small spaces, perhaps wrapped up entirely in quantum uncertainty. String theory is controversial. Some physicists damn the whole idea by declaring it "not even wrong."

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The above title is rather redundant. Geometry literally means "earth measure"; but as first developed by Euclid of Alexandria, its focus was on the descriptions of geometrical shapes that could be accommodated to a flat surface, such as a sheet of paper. The Earth certainly seems flat to us small creatures that crawl like ants over its surface, but as humanities activities have increased over its surface, through the school of hard knocks people have learned that Euclid' s description of geometrical relationships over the Earth's surface was too limited and therefore required somegeneralized.

As humans explored more of the Earth, in the process of enlarging their knowledge of distant lands and widening their trades, they found that the geometry of Euclid was inadequate to describe the relationships of large geometrical figures over the surface of the Earth. Some of the great thinkers among the Greeks believed that the Earth was a sphere and not flat. In Alexandria, Egypt, the Greek scholar and philosopher, Eratosthenes (276 BC – 195 BC), came close to measuring the radius of the Earth based on the angles of shadows cast by the sun in Egypt on the Summer solstice. The model that he used for this purpose was that of a spherical earth embedded in a flat, three dimensional space.

The idea that the Earth is nearly spherical was forgotten somehow in ancient times, but was regained in more modern times. In exploring large portions of the surface of our planet, map makers became aware that you just could not faithfully represent large portions of that surface on a flat sheet of paper without distortions. The way to get the geometry right was to draw it on a sphere. In those days, the map of the Earth on globes in someone's study became the mark of an well educated person.

Confronted by the reality of living on an almost spherical planet, mathematicians were almost forced to generalize Euclid' s plain geometry to that of a spherically curved space. The two dimensional surface of the planet Earth is not flat. It is a curved surface, which is very nearly a spherical surface. Although horizontal surfaces on the Earth seem quite flat to us, they are really small sections of spherical surfaces, which to us seem quite flat. Mathematically, the surface of a sphere can be described as a two dimensional manifold.

There are direct confirmations that the Earth has a spheroidal shape. The Earth's shadow cast on the moon during a lunar eclipse shows that its edge is circular. One of Galileo's early observations using a telescope was that sailing ships coming into harbor from far away were partially hidden from sight by the curvature of the earth's blocking light, which travelled very close to a straight line to the observer's eye. On his home-made telescope Galileo could see a the image of a boat approaching harbor, first only as the top of its sails, and progressively through the rest of it rising above the horizon.

The geometry of a two dimensional spherical surface is quite different from that of a flat plain. The curved space space of the earth's surface had to be handled a special way, as was discovered by the early map makers.

For convenience, navigators wanted maps that they could lay out flat on their desks. Unfortunately, reality did not allow the map makers to make flat maps of the earth without stretching or tearing the surface. They used map projections, which gave fairly accurate representations of the earth's surface on a flat map by allowing them to have systematic distortions. It became even possible to do accurate calculations on a flat map of a curved surface, by treating the distortions as functions of the map coordinates, which are called map scale factors.

]]>Some people trust only what they can touch and see. They want their whole concept of reality to be grounded on the familiar and the tangible. They imagine that such a picture of reality is solidly grounded; but, physics itself destroys such notions.

Physics began with what people saw around them, which is everything material that is contained in space and lasts long enough to be observed and measured. And yet, there is constant change. Physical reality is like a river. Hereclitus, the ancient Greek philosopher said " You could not step twice into the same river; for other waters are ever flowing on to you." Amid that change, what is permanent and what is unchanging? Many philosophers did not like the answer that some philosophers gave, "God is the unmoved mover of everything."

Unfortunately, the idea of God transcends all natural forms and cannot by His very nature be defined. A definition by its very nature distinguishes something from its surroundings. In a sense, a definition makes the concept of something finite by reducing it to a subset of something else. But what if you try to define the concept of God, who is infinite? Because the concept of God is so recondite, some philosophers preferred to find causes for everything to be within nature, which they could discover and study in terms of nature itself. The problem here is that nature is an emergent phenomenon and not an ultimate reality. It had had a beginning in an event that is called "The Big Bang". In the early moment of that creative event, nothing familiar existed. From that highly energetic event, "fundamental particles" of physics emerged. As these particles cooled they conglomerated into various structures, from which more familiar forms of matter emerged; and from these forms of matter, emerged the visible universe.

The early state of the universe and the emergence of mass, space, time and the four fundamental forces can be approached by considering the various states of matter. Changes that mark the passage of time are transitions that happen among all available states according to the equations of motion. Although the universe had its inception 13.8 billion years ago in our reckoning of its passage, time as it appears in the fundamental equations of motion, has no preferred direction. The equations of motion in physics are invariant under time-reversal (when -t replaces t); however, from thermodynamics, which concerns the behavior of microscopic constituents in the large aggregates that characterize matter on the large scale, emerge the concepts of thermally driven information flow and entropy, from which emerges the arrow of time of our experience. Time, as we feel it, cannot therefore be a fundamental property of physics, but rather, is an emergent property, which is produced by thermodynamics.

Not only is our perceived time an emergent property, but Big Bang Theory presents matter itself as an emergent property.

Einstein's famous equation, E = m c** ^{2}**, expresses the idea that mass is a form of energy that in fact, mass and energy are equivalent. Originally, everything that we see around us emerged from much more primitive states of energy in the Big Bang through a complex of physical reactions.

Consider the three common phases of ordinary matter: gas, liquid and solid. These three phases relate to the states of substances in response to changes in pressure, volume and temperature in ordinary Earth environments. Density and volume are closely related, and one can just as easily specify the states of matter in terms of pressure, density and temperature. Through a specific range of temperatures, densities and pressures many substances are gases, our own atmosphere for example is a gas at ordinary temperatures. At certain threshold temperatures, densities and pressures, gases condense into liquids; and beyond those, into solids. Generally, liquids are denser than gases, and solids, denser than liquids. Water is an important exception, note that ice floats on water.

The success of explaining the properties of substances in terms of atoms and molecules in motion ( kinetic theory ) was in being able to explain the phases of matter in terms of particles moving around in otherwise empty space. The properties of a gas, which exerts a pressure and can expand to occupy the volume equal to that of its container, are explained in terms of molecules of a gas bouncing elastically off each other and off the walls of a container. Likewise, the properties of a liquid, which occupies a fixed volume but assumes the shape of he containing vessel, are explained in terms of molecules that have been slowed down enough (cooled) and brought together close enough to stick under weak forces between them, so that they still tumble over each other, but do not separate. Further cooling allows molecules to freeze in place into some regular geometric pattern, or crystal, clusters which constitute a solid.

In kinetic theory, the temperature of a gas is a measure of the average kinetic energy of a molecule, which in a gas, moves freely until it collides with, either other molecules, or the walls of the container. Something similar happens in a liquid, but in a more restricted way. Molecules move around each other while still being held in close proximity. In a solid, the molecules vibrate in place about their average position. Kinetic theory works for gases as long as the approximation of particles moving in a void works. In reality, there is no true void. Space is a subtle medium that is full of zero-point energy, which manifests as a plethora of fluctuations including electromagnetic ones. Although space is not nothing, it is not ordinary substance that occupies space; there is no *"there"* there.

A gas breaks down at some temperature when the average energy of each molecule becomes greater than or equal to its binding energy. Beyond this threshold, the thermal pounding of molecules against each other is strong enough to break the bond between the component atoms, in which case molecules are replaced by their more fundamental constituents, which are atoms that contain an excess or deficit of electrons. The gas then becomes a mass of charged energetic particles that form what is known as a plasma.

The particles of physics are not bits of matter at all. Instead, we think of them more as processes, such as whirl winds or clouds. The reality of molecules emerges from their states of motion, which are defined over a restricted energy range only. Within that range, the particles behave like things that have an integrity, but beyond that range the particles resolve into their constituents, which may also behave like particles.

At a high enough temperature the molecules of a gas break down into a gas consisting of charged bits of the constituent atoms, which occasionally recombine into molecules. Individual molecules have a lifetime that is limited by the energy of random collisions of other components of the gas. Increasing the temperature even further, so that the average kinetic energy of the gas components passes the binding energy of the orbital electrons of the atoms of the gas, begins to compromise the integrity of the atoms themselves. The result of this extreme condition, is that now charged, subatomic particles appear among the gas constituents, now forming a plasma. Further increases in temperature result in atomic nuclei's further disintegration into protons and neutrons. The series continues, as temperature and pressure are increased, at some point causing electrons to recombine with protons to form neutrons, which is observed to have happened in a neutron star. The eventual result of this process is the compression of matter into quark gluon plasma, which are the components of a heavy particle of the atomic nucleus, or *hadron*. Ultimately the collapse, results in a black hole. In the end, gravity wins over all, remaining behind, like the Cheshire Cat's smile, when everything else (except information) has been sucked in and disappeared behind the black hole event horizon.

Although we cannot draw up a definition of Divine operations in nature and therefore we are unable to incorporate Divine operations into the laws of physics, it is possible to philosophically discuss the connection of an Infinite Creator with the product of that creation in nature. Some of this discussion was taken up in a previous post, Substance and Form–Philosophy and Physics, in which some of the historical philosophical material on the subject was reviewed.

In his book, *The Republic*, Plato himself recognized that the world of our senses is a kind of deceptive world that shadows a deeper reality. He expressed this relationship through the allegory of the cave, in which people say reality in terms of its shadows cast on a wall of a cave. But what is that reality? In the Judeo-Christian tradition, that reality is the Infinite God Himself, who is the Creator.

Indeed, there is some esoteric Jewish philosophy, the **Kabbalah** that attempts establish some connection between the Infinite, unknowable God with his knowable creation.

That idea is further elaborated by Emmanuel Swedenborg, the Swedish philosopher, scientist and mystic, who stated that all of creation is built up of several discrete levels, the articulation of which rhyme by a precess he calls correspondence. The law of Correspondences is a key concept of Swedenborg's theology and philosophy that links the whole of creation to the infinite, eternal God, who is ultimate reality and the final cause of creation. The concept of correspondence is a powerful, philosophical tool which has a broad base in natural science, where there is a lot of self-similarity and fractal structure that repeats on many scales.

]]>Astronomers operating the space telescope Kepler have detected a large mega-structure orbiting a star 1500 light years away, which some suggest may be a huge orbiting energy collector for what would be a very advanced civilization of sentient beings—a Dyson Sphere structure! A recent article appearing on the Internet states the following:

A paper has been submitted to the journal Monthly Notices of the Royal Astronomical Society in which a particular star named KIC 8462852 is described.

“Over the duration of the Kepler mission, KIC 8462852 was observed to undergo irregulary shaped, aperiodic dips in flux down to below the twenty percent level. We’d never seen anything like this star, it was really weird. We thought it might be bad data or movement on the spacecraft, but everything checked out.” –Tabetha Boyajian, researcher at Yale University

The above quoted article is based on an article appearing in the Atlantic entitled, "NASA’s Kepler Telescope Discovered Artificial Megastructure Built By An Advanced Alien Civilization".

On another website (wccftech) the discovery of another potential Dyson Sphere is posted in an article entitled, "A Second ‘Dyson Sphere’ Star Has Been Discovered – And We Have An Alternative Hypothesis To Explain Them". Another article, "Search For Intelligent Aliens Near Bizarre Dimming Star Has Begun", indicates that the mysterious star system containing a potential Dyson Sphere is being probed by the Allen Radio Telescope array near San Francisco.

Kurzweil devotes a small article ["Dyson sphere hunt using Kepler data"] to the discovery of this as as a potential Dyson Sphere.

If indeed a Dyson Sphere has been detected then some sentient beings in the universe have achieved a much higher level of civilization than our own. How do we recognize the various levels of civilization? How can we say that a particular civilization on another planet on another stellar system is more advanced than ours?

The Russian astronomer Nikolai Kardashev in 1964 proposed a three level scale to measure levels of possible civilizations of humanoids and other sentient beings of the universe. This scale, called the Kardashev scale, simply measures planetary economies by the amount of energy they consume.

On the planet Earth, we have a Type 1, or planetary civilization, in which we use energy that reaches us from our parent star, the Sun. Fossil fuels are a form of stored solar energy so that burning coal or using gasoline is tantamount to using energy from the Sun.

As dependant as we are on the energy of the sun, we receive but a minuscule portion of its output. The Sun's energy output is estimated at 384.6 yotta Watts (1 yotta W = 1 x 10^{24} J s^{-1}). See post on Energy Units. The amount of solar energy radiation arriving at the top of the atmosphere is about 1,368 W m^{-2} (Watts per square meter). This radiation spread out over the surface of the earth and averaged out over a diurnal cycle amounts to a total power of 1.730×10^{17} W. This is a lot of power reaching us. One year is 3.15×10^{7} seconds. Multiplying this value of a year in seconds by the amount of power reaching from the Sun amounts to 5.46×10^{24}J. The International Energy Agency estimates that, in 2013, total world energy consumption was 5.67 × 10^{20}J. This is just about 0.010 % the total solar energy input for a year.

A few caveats should be added regarding the use of energy consumption as a measure of the level of a civilization. The per capita energy consumption is quite variable over the Earth as is apparent from this article and this figure, from which the displayed image (Fig. 1) is drawn. This image shows how variable the level energy consumption is of planet wide. It does seem to correlate with our ideas of the various levels of civilization around the globe. However, the raw use of energy is not a unique measure of the social progress of a given society, which depends on other factors (see the article and other image on Per Capita Energy Consumption vs. Social Progress). Just keeping warm demands a lot of energy for people living near the north and south poles, which may not necessarily contribute to their level of civilization. Further, different areas of the world use energy more efficiently or wisely than others.

On the next level of potential cosmic civilizations, Type II on the Kardashev scale, sentient beings capture all of the output of their parent star, by building structures around it to increase the fraction of total energy output. A structure that completely surrounds their parent star is called a Dyson Sphere. By using a Dyson Sphere, a civilization can expand on a much bigger scale than that afforded by the limited area of the Earth. The Earth's surface is 4πRe^{2} , where Re=6.378 x 10^{6} m is the Earth's radius. The inside surface of a sphere that has a radius equal to the Earth's distance from the Sun is 4πau^{2} , where 1 au=0.1496 x 10^{12} m. The inside surface of a Dyson Sphere would exceed that of the surface of the Earth by a factor of (1au/Re)^{2} = 5.502 x 10^{8} times!

A Type III civilization taps into energy on the scale of its entire host galaxy! Beyond that, after harnessing black holes, there is the energy of expansion of the entire universe, and that is really huge!

Many will say that all this is interesting but a bit too far out for practical consideration. Interesting SciFi stories can be written around these concepts: see for example the Star Trek episode "Relics", or Isaac Asimov's galactic civilization in the Foundation Series. But wait! Reality is strange and wonderful. What appears as today's SciFi may become tomorrow's science.

The possibility that there may be higher order civilizations out there in the universe, have caught my imagination, as a result of which I have decided to write a series of posts discussing the matter. Look out for such articles in the future. What do you think?

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