The Greek astronomer Aristarchus of Samos proposed more than two thousand years ago the first known model that placed the Sun at the center of the known universe with the Earth revolving around it. His description of reality has in the course of the last five centuries been further developed by Nicolaus Copernicus, Johannes Kepler, Isaac Newton, Albert Einstein and many others. Today, it is interesting for the millions to grasp the cosmic proportions of The Solar System and also of the Milky Way galaxy in which the Sun is one of more than a hundred thousand million stars and which is merely one of the same number of galaxies in the universe. Another matter of great interest is the indication from observational data that all the galaxies are affected by some kind of Dark Energy and Dark Matter. Their nature is discussed and there are many speculations but no generally accepted theory.

Descriptions follow below of four handheld cosmic models showing the Solar System and the Milky Way as well as of an explanatory model proposed for Dark Energy and Dark Matter. These models are the first and only of their kind. They are called Demonstrators and intended to be manufactured as square plates adapted to communicate with an optional electronic device for user interaction.

Demonstrator 1A is a Four-Planet Model of the Solar System provided on a front side of a square plate. The sizes of Mercurius (Mercury), Venus, Tellus (Earth) and Mars are at the corners of the plate shown in proportion to the size of the Sun as represented by the outer circle. This circle is also used to represent the orbit of Mars. The inner circles represent the orbits of Tellus, Venus and Mercurius. The scale along the top of the plate shows that light takes about 500 seconds to travel from the Sun to Tellus.

In proportion to the size of the planetary orbits which have the shape of ellipses approximated by circles, the size of the central Sun is shown as well as the eccentricities of the orbits. While these are for Mercurius and Mars quite noticeable, the circular shape is a close approximation of the elliptical orbits.

On the back side of the square plate, Demonstrator 1B is an Eight-Planet Model of the Solar System. The sizes of Jupiter, Saturnus (Saturn), Uranus and Neptunus (Neptune) are at the corners of the plate shown in proportion to the size of the Sun as represented by the outer circle. This circle has the same size as the outer circle on the front side and it is also used to represent the orbit of Neptunus. The inner circles represent the orbits of Uranus, Saturnus and Jupiter. The scale along the top of the plate shows that light takes about 4 hours to travel from the Sun to Neptunus.

Noticeable eccentricities of the planetary orbits are shown as well as the size of the central Sun in proportion to the size of the orbits. The Asteroid Belt is shown between the orbit of Jupiter and the orbit of Mars in the inner portion of the Solar System. The latter is in Demonstrator 1A magnified 20 times.

From Earth, the Moon seems even larger than the Sun although it is somewhat smaller than Mercurius. The Sun and the Moon have roughly the same diameter ratio as distance ratio. The distance to the Moon equals 30 times the diameter of Earth or, as a remarkable fact, the summed diameters of Earth's seven fellow planets Jupiter, Saturn, Uranus, Neptunus, Mars, Venus and Mercurius. 

It was decided in 2006 by the International Astronomical Union that The Solar System has eight planets. Previously viewed as the ninth planet, Pluto revolves outside the scope of Demonstrator 1B in an orbit together with plenty of other bodies. The total mass of these is much larger so Pluto is called a dwarf planet. However, also the ordinary planets resemble in the big perspective of the entire Solar System merely dust circulating in the air and reflecting light from a lamp. 

Demonstrator 2A shows on a front side of a second square plate the Milky Way galaxy - our cosmic home town - in a generally accepted interpretation of observational data (Credit: NASA). The universe is estimated to have some hundred thousand million galaxies in various sizes and shapes. The scale along the top of the plate shows that light takes about a hundred thousand years to cross the Milky Way from one side to another. Our galaxy has a central black hole, a galactic bar, and a few spiral arms. 

The Milky Way is called a barred spiral galaxy. The Solar System is located at the Orion arm and revolves the central black hole with an orbital period of about two hundred million years and with an orbit radius of about thirty thousand light-years. Observational data indicate that the galaxy is wrapped in a cocoon of invisible Dark Matter holding it together and making all the stars move in their orbits at roughly one and the same speed regardless of the orbit radius, and that all the galaxies in the universe are acted upon by something called Dark Energy causing an accelerated expansion of the distances between them.

On the back side of the second square plate, Demonstrator 2B is a graph that shows an explanatory model for Dark Energy and Dark Matter. It is a generally accepted view that almost fourteen billion years ago the universe had a start 1 of expansion called the Big Bang, that the expansion is related to a variation in the metric of space and has started from nothing, and that this expansion has a positive acceleration at the present point of time 2 as shown by the graph rotated 90 degrees clockwise. Some scientists predict a prolonged phase of accelerated expansion 3 until all the elements of the universe break apart. Other scientists propose that quite soon the positive acceleration could cease and turn negative resulting in a phase of contraction 4.        

The expansion and contraction phases 3 and 4 are marked with arrows between which a mathematical model is shown. The model describes the shift from positive to negative acceleration by means of exponential expressions using the natural base e raised to the power of x and to the power of -x. The contraction results in an end 5 as shown by the graph rotated another 90 degrees clockwise. A hasty interpretation would be that the birth of the universe in the Big Bang is followed by its death with everything disappearing in a Big Crunch.

It is a fact, however, that the above-mentioned mathematical model is derivable from a most fundamental mathematical model of repetitive phenomena, namely the sine wave model. Rotating the graph another 90 degrees clockwise, the shape of the sine wave is shown by the dashed line. If the variation in the metric of space is expressed as a ratio between a measure dt_c of cosmic time t_c and a measure dt_r of reference time t_r where dt_r is proportional to the second defined by atomic reference clocks and dt_c is proportional to the metric of space as determined by the constant speed c of light, then the equation dt_c/dt_r = sin (2π t_c/T) makes x in the above-mentioned exponential expressions equal to 2π t_r/T where T is the repetition period of the sine wave. 

In the graph, the end 5 is followed by a new start in a phase that is shifted 180 degrees with respect to the start 1 and that results in new expansion and contraction phases. The two mathematical models have a common peak 6. According to the equation above, the measure dt_c alternates beween positive and negative values so the cosmic time t_c flows in alternating directions inside the dashed line. Curved spacetime describes gravitation and the primary cause behind the curvature can be interpreted by describing simply the cosmic time t_c as flowing in a closed loop. Its repetition period T determines the mass of omnipresent elementary particles of gravitation which provide an interpretation of Dark Matter. As to the positive acceleration in the expansion of space, it is here derived from the sine wave model without any assumption of Dark Energy as dominating in the universe and causing a stronger force than gravitation. That assumption about Dark Energy can one day be viewed as the cause of the greatest bubble in the history of science. 

Like the hasty summary "Everything is relative" in the pop version of Einstein's theory, it can be said that "Repetition rules everything". Most things revolve and rotate in microcosmic as well as in macrocosmic structures. The ratio between the electromagnetic and gravitational forces results from a repetition period ratio of closed loop time flows for the reference time t_r and the cosmic time t_c determining a mass ratio between the elementary charge and an elementary particle of gravitation.

There are two half-periods in the graph. It is of no importance neither in which one of them we are at the present phase of cosmic expansion nor if parallel universes exist in the same or opposite phases. The possibility to describe opposite polarities of electric charges as opposite phases of a closed loop time flow is, however, of interest to discuss. Nevertheless, the explanatory model offered by Demonstrator 2B challenges all the present interpretations of Dark Energy and Dark Matter so this portion of the text as well as the graph on the back side of the second square plate will in the manufactured product be hidden by a black cover provided with a warning to the reader that the cover is removed at his or her own risk... 

Reference: "A New Cosmic Perspective" published by the Swedish Mathematical Society in the fall of the International Year of Astronomy 2009.