Polytopes: Part I

Polytopes, simply defined, are geometrical shapes. They can exist in any dimension, and are usually finite. For the purpose of the post, all polytopes are considered finite unless specifically designated to the contrary.

Starting in zero dimensions, there is no direction, and the only thing that can exist is an infinitesimal speck, called a point. In one dimension, an infinite number of distinct points exist. By connecting any two of these points, a line segment is formed.



A point and a line representing zero and one dimensions.

In abstract mathematical space the line must continue forever, representing the entire one-dimensional universe.

In two dimensions, there are an infinite number of lines possible, all crisscrossing what is called a plane. We are familiar with many shapes, called polygons, on this plane. One and two sided polygons aren't really possible with straight line segments, but one and two sided polygons with arcs of a circle as sides, called henagons and digons are technically possible. The simplest real polygon however, is the triangle. A triangle is the shape defined by the intersections of three distinct nonparallel line segments. A picture of a triangle is shown below.



A triangle. The line segments are called the sides, and each pair of sides marks an angle. The triangle is the interior region defined by three line segments, and including the line segments. The rest of the plane is known as the exterior.

For a triangle, if it is equilateral, or all the sides are the same length, it follows that the triangle is also equiangular, or that the angles all have the same measure. This only follows for the triangle and not any higher polygon. The triangle that is both equiangular and equilateral is called a regular triangle. It is given the symbol {3}. The angles of any triangle in regular space must add up to 180º, with a general Euclidean space polygon's angles add up to 180º(p-2) where p is the number of sides in the polygon.

Moving on, one can define a polygon with four sides, known as a quadrilateral. From the formula mentioned above, it is clear that the angles of a quadrilateral add up to 360º. With four sides, a polygon can be equilateral without be equiangular, the example being the rhombus, and can also be equiangular without being equilateral, the example being the rectangle. The two shapes are shown below, along with the quadrilateral that is both equilateral and equiangular, the square, which is denoted by {4}.



All of the above polygons mentioned are convex, meaning that none of their angles' measures exceed 180º. However, with polygons of four sides and up, concave polygons exist, and these can even intersect themselves. When a polygon intersects itself on the plane, it is known as a complex polygon.




The above is a concave quadrilateral, followed by a complex one.

In two dimensions, a polygon with any number of sides can exist and the regular convex pentagon {5}, regular convex hexagon {6}, regular convex heptagon {7}... etc. follow the quadrilateral. In general a regular convex polygon {p} has p points, or vertices, and p line segments, or sides. The total angular measure of a regular convex polygon is 180º(p-2), and each individual angle is 180(1-2/p), however, the sum of exterior angles of any regular convex polygon is always 360º.

A polygon with p sides can also be generated by taking the pth roots of one, and plotting them on the complex plane. Each vertices' coordinates is found with (cos(2πik/p),sin(2πik/p), where k is {0,1,2,3...p-1}. The pentagon found be connecting the 5th roots of one is shown below



Note that all five points lie on the unit circle, or the circle with radius one centered at the origin (the point (0,0)).

However, a regular polygon does not have to be convex. There are regular polygons which are complex. These are known as star polygons. Each star polygon with p sides has the same vertices as a regular convex polygon {p}, but the sides are connected in a different way. The simplest star polygon is known as the pentagram, and is shown below.



This particular star polygon is denoted {5/2} because the points are connected in such a way that every point is connected to the point two points away from it in a counterclockwise order. With five sides, {5/1} simply connects the points in order, simply resulting in {5}, {5/3} is a simple reflection of {5/2} and {5/4} is just the points connected in the opposite order, still resulting in {5}.

In the case of {6}, no continuous star polygon can exist, because {6/1} and {6/5} trivially equal {6}, and the remaining three, {6/2}, {6/3} and {6/4} are not continuous, meaning that despite what point of the hexagon one starts at, not all the points are accessed, and the polygon is made up of multiple parts, two for {6/2} (the Jewish star) and {6/4}, for which the parts are triangles, and three for {6/3}, the parts being lines.



The star polygon {6/2}. It is made up of two disconnected triangles, and is therefore not a true star polygon.

The number of distinct continuous star polygons, disregarding simple reflections and rotations, for all p up to 10 are

p=1-4: 0
p=5: 1; {5/2}
p=6: 0
p=7: 2 {7/2}, {7/3}
p=8: 1; {8/3}
p=9 2; {9/2}, {9/4}
p=10 1; {10/3}

All possible star polygons with p up to 16 are shown below (note p is replaced by n in the picture).



In the above picture, the red diagonal lines represent {p/1} which is simply {p}, {p/2}, {p/3} and so on and the blue diagonal lines show the common names for these star polygons. Note that the continuous star polygons are in boldface, while the discontinuous ones are not (click on the image to enlarge it).

In conclusion, a star polygon {p/x} is continuous when GCD(p,x)=1 (greatest common divisor of x and p). An equivalent statement is that p and x are relatively prime. GCD(p,x) is also the number of discontinuous elements required to include all the points. For example, since GCD(5,2)=1, {5/2} is a continuous star polygon, but GCD(6,2)=2 and two distinct pieces (in this case triangles) are required to include all the points with a {6/2} polygon. Additionally, if p-x=±1, then {p/x} is equivalent to {p} and is a special case of the more general star polygon, the convex polygon.

A final notable case of polygon is the star-shaped polygon, which is very different from a star polygon. For any star-shaped polygon, there exists one or more points on the interior of the polygon from which a line can be drawn to each vertex without intersecting any other part of the polygon. An example of a star polygon is shown below.



The top image shows how the polygon's interior contains at least one point from which all vertices are visible, or accessible with lines without obstruction. In addition, the set of points where this is true is named with the (quite odd) name, the kernel. The lower image shows the kernel of this particular star-shaped polygon. Complex polygons cannot be star-shaped, and, ironically, nor can star polygons. Convex polygons are all star-shaped, and their kernels coincide with the polygons themselves. Additionally, all triangles are star-shaped by definition, because they all are convex.

This post is continued in Polytopes Part II.

Sources: Regular Polytopes by H.S.M. Coxeter, Google Images

Orbital Resonance

An orbital resonance is a gravitational phenomenon in which two bodies that are both orbiting around one parent body are in a specific pattern. For example, if for every orbit Planet Bill makes around a star, Planet Joe makes exactly three, the two bodies are in an orbital resonance (the notation for this is 1:3). It may seem unlikely that such a correspondence would occur in the physical universe and that this is a purely abstract notion, but this is not the case. In fact, gravity often pushes objects into these resonances when the objects are of comparable size, but ejects them if one is too much bigger than the other, as with Jupiter and an asteroid.

In fact, there are resonances in our Solar System! The simplest case is the one between Neptune and Pluto. For every two orbits Pluto makes, Neptune makes exactly three. This is why it is impossible for the two bodies every to collide, despite the fact that their orbits cross. This is the only stable resonance involving two planetary (although Pluto is a dwarf planet) bodies, but other objects outside of Neptune's orbit are in resonances with Neptune, known as Trans-Neptunian objects. Most of these are smaller than Pluto, but one larger one is known: the dwarf planet Eris. However, Eris is not known to be in an exact orbital resonance with Neptune.

Other Trans-Neptunian objects sometimes are in resonances with Neptune, the most common being (object:Neptune) 2:3, corresponding to Pluto and other bodies, 3:5, 4:7, 1:2, and other rarer ones such as 2:5, 3:4, 4:5, 1:4, 1:5, 1:3, 3:7, and 6:11. Some of the latter sometimes correspond to only one known object, and may be coincidental. Some are also unstable, and smaller objects can often be ejected from a resonance by a gravitational pull quite easily. The special case of a 1:1 orbital resonance is addressed in the post, Lagrangian Points.



A chart showing known objects, resonances, and distances of various objects beyond Neptune's orbit.

Resonances of a different kind impact asteroids in the main asteroid belt. Jupiter's gravitational pull has a great effect on the asteroid belt, and unlike Neptune's resonances, are much closer to each other. Therefore, rather than objects commonly existing at these resonances, repeated encounters with Jupiter ejects the asteroids onto another orbit. Due to this, there are gaps, called Kirkwood Gaps, that exist at the main orbital resonances with Jupiter, named after Daniel Kirkwood, who observed and explained the nature of the gaps in 1857. The population of asteroids in relation to the gaps is shown in the image below.



This image shows four main resonances and the effect they have on asteroid population. There are other weaker ones that cause a lower number of asteroids to keep stable orbits, but they are no nearly as drastic. Two examples are 7:3, shown at 2.71 AU, and 9:4, at 3.03 AU.

There are a few other major resonances in our Solar System, including a 1:2 between Saturn's moons Dione and Enceladus and the 3:4 one between Saturn's moons Hyperion and Titan. However, the most famous resonance in our Solar System is the only known Laplace resonance, or one involving more than two bodies.

It is the 1:2:4 resonance between three of the four Galilean moons of Jupiter. Io is the 1, Europa the 2, and Ganymede the 4. This remarkable property supposedly emerged with a gravitational encounter of Ganymede and another body, resulting with the instability of Ganymede's orbit. Due to the gravitational pull of Io and Europa coupled with that of Jupiter, Ganymede settled into its position in the resonance.

Outside of the Solar System, many other orbital resonances probably exist, between exosolar planets, and one known example is a pair of planets orbiting the star cataloged as GJ 876. They are comparable in size to Jupiter, and both orbit extremely close to their parent star, within what would be the orbit of Mercury, but have a 1:2 resonance. This type of resonance among major, Jupiter-sized planets is more significant than any other one known, and is unique, as far as we know.

There are also other types of resonances, such as secular resonance, which is the alignment of the precessions of two bodies. Precession is the cycle in which the axial tilt of a body orbits around back to its starting position. For the Earth, the axial tilt is 23.44º, and it takes approximately 41,000 years to run a full cycle. During this time, the axial tilt varies from 22º to 24º. Where the north pole points in space is known as the celestial north pole. Over time, this pole moves over the heavens, and the north star therefore changes. Currently, the closest major star is Polaris, but over the next tens of thousands years it will progressively move away and then back when the cycle is complete. As a result, Vega will also be the north pole star for a time in each cycle.

This and other types of resonance are common throughout the Solar System within seemingly complex and random gravitational interactions.

Another type of synchronization that may be called a resonance, is the tidal force of a satellite on a parent body. Over time, the pair of bodies tug on each other gravitationally, pushing the slightly heavier side of each inward so that the period of rotational of the smaller body is eventually exactly the same as its revolution period, and is tidally locked. Many moons in the Solar System exhibit this feature and the most well known is our own moon. We only see one face of it because it is tidally locked to the Earth. In addition to this however, there also is a gravitational tug from the moon that is slowly tidally locking the Earth and the Earth day is slowly getting longer, increasing by mere fractions of a second each year. In addition, the two moons of Mars, at least eight of Jupiter's moons, fifteen of Saturn's moons, five of Uranus, two of Neptune, and the Pluto-Charon system (both bodies are locked to each other). Also, there is one known extrasolar instance of mutual tidal locking in which a star and its giant planet are tidally locked to each other.

One other special type of tidal locking is the Mercury-Sun system. For every two revolutions of Mercury around the Sun, Mercury revolves thrice on its axis. This relation is known to be stable, but it is a unique case.

Resonances are mysterious connections in the heavens which are difficult to find and even more difficult to understand.

Sources: http://www.cfa.harvard.edu/mpec/K08/K08S05.html, http://upload.wikimedia.org/wikipedia/commons/1/1c/TheResonantTNO_90AU.svg, http://adsabs.harvard.edu/abs/1983Natur.301..201D, NASA, http://adsabs.harvard.edu/abs/1991Icar...94..399M

Lagrange Points

As of 2006, one condition specifying an object as a planet is that the object has "cleared its neighborhood". This refers to bodies large enough to gravitationally fling objects out of their orbits. Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune have all done that. Pluto and its Kuiper belt counterparts, known as Plutinos, often share orbits with one another, ending up in a jumble of asteroids. It is easy to see, by this definition, that Pluto (and other dwarf planets such as Eris and Ceres) is clearly not a planet. However, the gas giants have not completely cleared their orbits. Mysterious "Trojan" asteroids follow them around their orbits. But how can this be?

The answer requires physics. There are actually five points within a system of to bodies in which an object can orbit in stability, despite being within or very near a larger object's orbit. These are known as Lagrange points, named after Joseph Lagrange. The discovery of these points occurred while Lagrange was exploring the famous three-bodies problem in 1772 (calculating a stable system for three bodies with their respective masses, speeds, and positions at any given time). Despite the simplicity of a two body system, an additional object makes the result very complex, and many scientists devoted their time to researching this problem. Contrary to previous belief, there are actually five points at which stability is attained (rather than three, which was the result found by Leonhard Euler 20 years previously). All objects at these points have a 1:1 (read "one to one") orbital resonance with the smaller of the two main bodies, meaning that for every orbit that it makes, an object at any Lagrange point makes one too. They are described in the picture below.



The five Lagrange points are illustrated above, along with a description below. They are denoted by L1, L2, L3, L4, and L5, while the arrows denote the forces acting on the areas around these points.

L1
If a line is drawn between the two massive bodies, L1 will lie on that line, in between the two, at the point where the gravitational pulls of each cancel each other out. In the case of the Earth-Sun system, this point is much closer to Earth, due to the fact that the Sun is much more massive. In most cases, a body orbiting closer to the Sun than Earth, i.e. an object at L1, would orbit faster and have a smaller revolution period. However, the gravitational pull of the Earth keeps an object at L1 from speeding up, and the object is literally held at that point all the way around Earth's orbit. It therefore has the same orbital period as the Earth, at one Earth year.

L2
L2 lies on the same line as L1, but is outside the Earth's orbit rather than inside it. Here, the force from the Sun would usually have such an object orbit slower than the Earth, but the Earth's gravity adds pull to the object, speeding it up, and keeping its year equal to one Earth year. Essentially, both L1 and L2 are points at which the forces of the Earth and Sun add or subtract to from exactly one Earth year.

L3
L3, as with L1 and L2, again lies on the line extended from between the two massive bodies, here the Earth and the Sun. This time, the body is opposite the Earth on its orbit, lying outside the Earth's orbit, but also closer to the Sun than Earth. But how can this be? The above statement appears to be a contradiction, but is actually possible due to the pull that the Earth has on the Sun. Both bodies actually orbit around the barycenter of the Earth and the Sun, which, despite being well inside the Sun, is not at the center. Because of this, the Sun is always slightly farther from the Earth than it would be for a less massive object, such as a body at L3, and a position satisfying both statements is therefore possible. At L3, the forces of the Earth and the Sun again balance, and the body has the exact same orbital period as the Earth.

L4 and L5
L4 and L5 are grouped together because they are very similar and both have the same properties. They are the most notable of the five Lagrange Points. If the Sun and the Earth form two points of two equilateral triangles, then L4 and L5 are located at the third point of each triangle. L4 orbits 60º ahead of Earth, and L5 orbits 60º behind. Due to the fact that the points are equidistant from the Earth and the Sun, they receive as much pull from the Sun as the Earth does, and they are kept stable by the Earth's gravity. However, the points L4 and L5 are stable if and only if the ratio of the larger body to the smaller one exceeds 25. This is true for the system of the Sun and any planet, as well as the Earth-Moon system. Otherwise, an object orbiting at these points will be perturbed and its orbit changed.

The five Lagrange Points are not only in physics, but have been observed. Some spacecraft have used the point as a base for observations. The Earth-Sun L1 point is best suited for observations of the Sun, due to the fact that the view from this point will never be disrupted by the Earth or the Moon. One spacecraft that has used this point to its advantage is SOHO (Solar and Heliospheric Observatory) launched in 1995. The Earth-Sun L2 point is ideal for deep space observations, because its view will never be disrupted by the Earth, the Moon, or the Sun. One spacecraft planned to take advantage of the L2 point is the James Webb Space Telescope, which is planned to be launched in 2014 as the successor of the famous Hubble Space Telescope. Here, no light from the Earth or Sun will damage the equipment, and an unhindered view of the wonders of deep space will be available. L3 does not have any specific use, but some people once believed that another planet similar to Earth, called "Counter-Earth" existed at this point. Since this point was never visible from Earth, the possibility was real until more sophisticated technology disproved this hypothesis. L4 and L5 are possible sights of space colonization, and a base could be set up in a stable orbit in one of these points, if partial or total evacuation of Earth ever became necessary. However, this does not seem imminently likely.

In a perfect system with only three bodies, the points L1, L2, and L3 are technically stable, but in a multiple body system like our own Solar System, an object staying at any of these points over a long period of time is impossible. The gravitational effect from the other planets will eventually disrupt any of these orbits. However, L4 and L5 are a very different story, indeed. They are much broader than the other three points, and perturbations will not always disrupt their orbits. Rather, a perturbation at these two points will change the orbit of the point slightly, increasing the eccentricity, but still keeping the objects in the range of the points. Due to this, there is matter floating around in these areas for both the Sun-Earth system and the Earth-Moon system. However, in both cases, the debris is simply interplanetary dust, and does not form large objects. In the outer planets, however, there are larger objects. Within the orbits of Jupiter and Neptune, there are many asteroids occupying these areas. The are referred to as Trojan asteroids. Also, two moons of Saturn, Tethys and Dione, have two smaller satellites each orbiting in perfect synchronization with them.

The Lagrange points are important for many uses, many of which could be important to humanity's survival in the future.

Sources:
http://www.esa.int/esaSC/SEMM17XJD1E_index_0.html
http://en.wikipedia.org/wiki/Lagrangian_point
http://en.wikipedia.org/wiki/Three-body_problem