Author Archives: thelision

A different Kind of Geometry

Introduction

So in class we’ve been talking a lot about non-Euclidean geometries recently. Now while both spherical and hyperbolic geometries are pretty different from Euclidean geometry, they all shared some things in common. One such thing was that they were continuous.  So instead of dealing with continuous geometries let’s talk about a finite geometry.

Definitions

The following definitions and theorems, for the most part, were taken from the ARML (American Regions Mathematics League) Novermber 1999 Power Contest titled Twenty-five Point Affine Geometry. There are some modifications.

Definition 1: A point is any letter from A to I. There are nine points in this initial geometry. The points are arranged in blocks as follow:

A B C
D E F
G H I

 

A E I
H C D
F G B

 

(Note that there is more than one arrangement for which all of the definitions would be satisfied)

Definition 2: A line is any row or column in one of the blocks. Therefore, a line contains 3 distinct points. There are only 12 lines and every point is on 4 different lines. For instance, DEF is a line and point E is on the lines DEF, BEH, AEI, and ECG. Note that lines do NOT have endpoints, but rather are cyclical in nature. So in the line ADG, after G we come back to A again.

Theorem 1: Given any two distinct points, there is one and only one line containing both of them. For example, given points E and A, only the line AEI contains them both.

Definition 3: two lines are parallel if they have no points in common. The points are arranged in the blocks so that parallel lines must lie in the same block and both be rows or both be columns.

Theorem 2: Given any two distinct lines, either they are parallel or they have only one point in common.

Theorem 3: Given a line and a point not on the line, there is one and only one line containing the point and parallel to the given line. For instance, given the line DEF and the point H, the line GHI is the only line that is both parallel to DEF and contains H.

Definition 4: Two lines are perpendicular if one of them is a row and the other a column in the same block. For instance, ABC and CFI are perpendicular and intersect, while line ABC and AEI intersect at point A but are not perpendicular.

Theorem 4: Through any point, there is one and only one line perpendicular to a given line. For example, given ABC and C, only the line CFI is perpendicular to ABC and contains C.

Definition 5: The distance between any two points is the least number of steps separating the points. Furthermore, since the lines do not have end points, it is important to note that, for instance, on the line ABC, the distance between A and B is 1, the distance between B and C is 1, and the distance between A and C is also 1 and not 2.  In other words, these lines can be thought of as cyclic. Furthermore row and column distances are not equivalent. In other words, the distance between A and B is not the same as the distance between A and D. The first is equal to 1, while the second is equal to 1’.

Definition 6: Point x is the midpoints of two points a and b if x is on the line containing points a and b and the distance between a and x is the same as the distance between x and b.

Note: two figures are considered different if they have a different set of points.

Definition 7: A triangle is a set of 3 non-collinear points.

Definition 8: A quadrilateral is a set of 4 points. Not 3 of which are collinear

Definition 9: A circle is the set of four points a given distance (called the radius) from a given point (called the center)

In relation to Euclidean geometry

Postulates

The first postulate, namely that a straight line segment can be drawn joining any two points, still holds up in this geometry. This is pretty evident from theorem 1. If you don’t believe that theorem, feel free to actually go through each and every case. It’s finite so you can actually do this.

The second postulate does not hold, namely that you can extend line segments indefinitely into straight lines, unless you apply some weird definitions. We did define our lines as not having end points (see definition 5), but as extending the line further would simply be looping the line around itself over and over it can’t really be extended indefinitely.

The third postulate, namely that given a straight line segment, a circle can be drawn having the segment as radius and one endpoint as center, does hold, provided that we create an intuitive definition of line segment, and that we modify our selection of distances to only include 1 and 1′. Since these are the only two distances within this geometry, we are still choosing from “all possible distances”, there just aren’t very many of them. The line segment will give us the distance which we call the radius and one of the endpoints which we will use at the center, from which we can construct the remainder of the circle.

The fourth postulate, that all right angles are congruent, doesn’t really hold. The issue here is that the term “angle” isn’t defined. We have 3 types of lines: those that intersect but are not perpendicular, those that are perpendicular, and those that are parallel. From this one could attempt to construct 3 classes of angles defined by the intersection types of these lines, but there wouldn’t necessarily be a good way of comparing the angles.

The parallel postulate does hold, but not quite directly. It hold in the sense that Playfair’s axiom, which is equivalent to the parallel postulate, holds, and is actually stated in theorem 3. However while the Playfair’s axiom does hold, Euclid’s 5th postulate, that is two lines are drawn which intersect a third in such a way that the sum of inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough, is once again problematic due to a lack of a definition for the term “angle”.

Other dimensions

So far we’ve stayed within the constraints of these 9 points and two blocks. This should be suspicious to you as the title of the ARML Power Contest was Twenty-five Point Affine Geometry. The original contest took place using 25 points, A through Y, as arranged below.

A B C D E
F G H I J
K L M N O
P Q R S T
U V W X Y

 

A I L T W
S V E H K
G O R U D
Y C F N Q
M P X B J

 

A X Q O H
R K I B Y
J C U S L
V T M F D
N G E W P

Ok so what changed from our earlier dimension to this one?

Well first off, the number of points on a line has been increased from 3 to 5. Furthermore instead of each point being either 1 or 1’ away from each other point, each point is now either 1, 2, 1’, or 2’ away from each other point. The total number of lines goes up to 30 and each point is on 6 lines. Finally, circles have a center and 6 other points instead of 4.

But what entirely new things did we gain by adding this additional dimension?

Well for one thing we now have pentagons! If we define pentagons as a set of 5 points no 3 of which are collinear, then we can in fact construct pentagons on the 25 point geometry, something which could not be done on the 9 point geometry. One instance of a pentagon in this new geometry is ABGFN.

Furthermore, you can also actually make hexagons, provided that we define a hexagon as a set of 6 points no 3 of which are collinear. One such example would be ABFGNX.

We can also introduce the following definitions:

Definition 10: A parabola is a the set of points equidistant from a given point (called the focus) and a given line (called the directrix)

Definition 11: A line tangent to a parabola intersects the parabola at only one point and is not perpendicular to the directrix.  Of the five points of a parabola, one is the vertex and the other four form two pairs of endpoints of a focal chord, which is a cord going through the focus.

Note that these parabolas did technically exist in the 9 point geometry; they just don’t mesh very well with the lines defined in definition 11. For instance, the parabola with a focus at E and with the directrix ADG is defined as the set of {F, I}, as both F and I are the same distance from ADG as they are from E. However, this means that this parabola has no focal chords and we can’t tell which point is the vertex. In the 25 point geometry, by contrast, one example of a parabola is the one with a focus at M and with the directrix of AFKPU. This parabola is defined as the set of {L, J, T} and {D, X}. All 5 of these points are the same distance from the line and from the focus, but what that distance actually is varies, unlike the parabolas in the 9 point geometry. Furthermore, it has a vertex at L and JMPX and TMFD are the focal chords. So while parabolas, from just the definition of a parabola, do exist in the 9 point geometry, they don’t interact very well with the lines associated with parabolas.

Generalized dimensions

We can actually create a geometry of this style for any (2n+1)2 number of points where n is an integer greater than or equal to 1. Since each point needs to be connected by a line to every other point, we can easily figure out how many square blocks we need to make this work. Let s be the number of square blocks. Then s*2*((2n+1)-1) =(2n+1)2-1, where (2n+1)2 is the total number of points.  This is because the total number of connections required per point is given by the right side while the left side is in the form of number of connections per block times the number of blocks. Thus s, the number of square blocks, is equal to n+1. It is also clear from this that you cannot create a geometry similar to these ones with (2n)2 points where n is any integer greater than or equal to 1 because the number of required square blocks would be given by

s*2*(2n-1)=(2n)2-1

where s and n represent the same things as before. Solving for s yields

s=(4n2-1)/(4n-2)

which is clearly not an integer as the top will always be an odd number while the bottom will always be an even number. This means that s will not take on an integer value which means that we need a non-integer number of square blocks, which doesn’t actually make sense.

Symmetry

Look back at the original 9 point geometry we defined. The point E is somewhat unique, in that it occupies the center of the first square. This is not true for any other points…or is it? Recall that lines don’t have end points. This means that if I go up from the point B in the first square, I end up at the point H. Going up from A will place me at G, and up from C will place me at I. You could, in fact, move that bottom row to above the current top row without actually changing anything about the geometry. In this new layout, B is in the center. This process also holds for left to right and so on, so anything that holds for one of the points, holds for all of the points. It may take some rearranging for this to become clear, but it’s definitely possible to do so.

Generalized Dimension properties

So obviously a geometry with (2n+1)2 points will have that many points, and will have the number of squares given by the formulas in the previous paragraph. Lines will contain 2n+1 points and each point will be in 2*s number of lines where s is the number of square blocks. The number of possible distance is given by 2n. A circle will be defined by the radius and 2*s points where s is the number of square blocks. The number of possible circles is given by the number of points multiplies by the total number of possible distances, which results in         (2n)*(2n+1)2 total circles.

To find the largest possible polygon, we use the following process. As a polygon must contain at least one point, we pick a point. The remaining number of points is now given by (2n+1)2 -1. As the polygon must be connected by lines, we will now choose a point that lies on the same line as the first point. Furthermore, as no three points can be collinear, by our definition of a polygon, the other points that were on this line can no longer be considered. Thus choosing this second point removes another 2n points from the remaining points. This means the total number of remaining points is now given by (2n+1)2-1-2n. Each time we choose another point we remove another 2n points from the remaining points. Thus, in general, remaining points    p=(2n+1)2-1-2nk where k is the number of points we have already chosen. By setting p=0, the least possible value p can take on while still making any sense, we can solve for k in any given geometry. So for instance, in our 9 point geometry, n=1, we end up with 0=9-1-2k,  from which we can clearly get that k=4, so the largest possible polygon in the 9 point geometry is a quadrilateral. In the 25 point geometry, n=2, we get 0=25-1-4k, from which we clearly get that k=6, so the largest possibly polygon in the 25 point geometry is a hexagon.

In general, we can solve 0=(2n+1)2-1-2nk to get k=2n+2. Thus the largest possible polygon in a given (2n+1)2 point geometry will have 2n+2 vertices.

Conclusion

In this finite affine (meaning that given two points there is only 1 line between them, Playfair’s axiom holds, and that it is possible to chose 4 points so that no 3 of them are colinear) geometry, there are many interesting properties. Some have been discussed here, like the largest possible polygon in given dimension, while others, like some properties of parabolas, have been left out. This is by no means a full exploration into the properties of this specific geometry, but rather a basic exploration into how some things within this geometry work. You should prove more things about it.

References:

Weisstein, Eric W. “Euclid’s Postulates.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/EuclidsPostulates.html

ARML (American Regions Mathematics League) Novermber 1999 Power Contest titled Twenty-five Point Affine

http://en.wikipedia.org/wiki/Finite_geometry

Vandalism and Mathematics

PROTIP: You can get around the Shannon-Hartley limit by setting your font size to 0.

Image: Randall Munroe.

Regarding the comic

This xkcd comic has two points.  The first is understandable without any context. If the writer had in fact discovered a proof that information is infinitely compressible, then ANY amount of space would be sufficient to contain it. The second point refers to Fermat’s famous statement “I have discovered a truly remarkable proof of this theorem which this margin is too small to contain,” which was, of course, referring to Fermat’s Last Theorem, a topic which we discussed extensively in class.

Liberal use of others property

It is now often believed that Fermat did not actually have a correct proof of this theorem. This minor detail did not, however, deter the great Fermat from writing it as fact in the margin of his copy Arithmetica to be discovered posthumously and baffle mathematicians for centuries to come. This, however, is not the only case of mathematicians writing statements in strange places. Another mathematician who did this was William Rowan Hamilton. Unlike Fermat, Hamilton decided to actually carve in his answer to a question, as opposed to carving in a claim that he has an answer. To be fair though, Fermat did own his book, while Hamilton didn’t actually own the bridge. This occurred in 1843, while taking a walk, he had a flash of brilliance during which he discovered Quaternions. Lacking a proper way with which to write down the result, Hamilton instead chose to carve his answer in the side of a bridge.

But what are quaternions?

Hamilton knew how to add and multiply complex numbers in a plane. However, he did not know how to multiply them in space. Quaternions were his solution to this problem, because while he could not figure out how to multiply complex points in a 3-dimensional space, he could figure out how to do it in a 4-dimensional space. In fact there is now a theorem which says the only normed division algebras which are number systems where we can add, subtract, multiply, and divide, and which have a norm satisfying |zw|=|z||w| have dimension 1, 2, 4, or 8. Quaternions can be thought of as a 4-dimensional space and are often denoted by H or ℍ. They are a noncommutative number system over the complex space, which just means that a*b does not necessarily equal b*a. They are defined as ℍ ={a+bi+cj+dk} where a, b, c, and d all belong to the real numbers. Note in particular that ij = k = -ji, jk = i = -kj, and ki = j = -jk. This eventually leads to what Hamilton engraved on the Brougham Bridge: i2 = j2 = k2 = ijk = -1, which means that ij, and are all equal to square root of -1.

Utility of quaternions

The quaternions can be used to do rotations in 3 dimensions, which may seem unintuitive given that quaternions describe a 4-dimensional space. To better explain this we need the concept of real and pure quaternions. A real quaternion is one which contains only a real part, while a pure quaternion is one which does not contain a real part. This is the equivalent of partioning a complex number into its real and imaginary parts. The difference between these two scenarios is that the pure portion of a quaternion is a vector in 3-space instead of a single number. Thus a real quaternion will take the form [a, 0] where 0 is the zero vector and a pure quaternion will take the form [0, v] where v is a vector of the form v=bi+cj+dj. Note that this means that the set of all pure quaternions define a 3-space. Thus the process of rotating in three dimensions is accomplished by starting with a pure quaternion, called p. This quaternion is then multiplied by the rotor, a second quaternion, called q, of the form [cos(Θ), sin(Θ)*v] where v  is a vector of the form v=bi+cj+dj and Θ is the angle by which we are rotating. If p happens to be perpendicular to q then the result will be a pure quaternion and the process is complete. However, if it is not the resulting quaternion will not be pure and the magnitude will be off. We can, however, multiply this new result by the inverse of q which will result in a pure quaternion of the desired length. Note that this means that the object should start and end in the 3-dimensional space as defined by the set of all pure quaternions with the real portion being used as an intermediary. I should also mention that the inverse is the conjugate of the quaternion divide by its normalization squared, where the conjugate is computed by negating the vector v and the normalization by dividing by the magnitude of the quaternion. Quaterions, however, don’t just allow for rotation in 3 dimensions, they also help avoid certain problems such as gimbal lock. Gimbal lock occurs when two out of the three rotational axes align. When this happens, the aligned axes both rotate the object in the same way. While you can still get out of the gimbal lock, it does force you to do some additional rotations. Quaternions circumvent this problem by having that intermediary 4th rotational axis.

Conclusion

If you want to commit vandalism, all you have to do is discover something brilliant which will be used for quite some time after its discovery in technologies which have yet to exist and engrave it in the side of a bridge or scribble it within the margins of a book. You might even get a plaque commemorating your vandalism.

Image: JP, via Wikimedia Commons.

Sources

http://math.ucr.edu/~huerta/introquaternions.pdf

http://www.cprogramming.com/tutorial/3d/quaternions.html

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/

http://en.wikipedia.org/wiki/Quaternion

http://www.gamedev.net/page/resources/_/technical/math-and-physics/do-we-really-need-quaternions-r1199

http://www.gamedev.net/topic/25314-why-diana-grubers-wrong-about-quats/

http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

http://3dgep.com/understanding-quaternions/

https://www.youtube.com/watch?v=zc8b2Jo7mno

A History of Mathematics, Uta C. Merzbach and Boyer.

Faith and Math: the Origins of Math

But to us there is but one God, plus or minus one. —1 Corinthians 8:6±2.

Religions. Image: Randall Munroe.

Religion and math are oft thought of as being separate and often in opposition, at least within western society. We recently learned about the connections between math and religion in India (http://www.bbc.co.uk/programmes/b03c2zvr) but did not explore where else faith has had an impact on mathematics.

Where does math come from?

The main two answers to this are as follows: humans discover math, or humans create math. In the case of the first, it is accepted that all of math exists, has existed, and will always exist, regardless of whether or not we are aware of it. Even though the ancient Greeks were unfamiliar with negative numbers, negative numbers existed, but had simply yet to be discovered. This mode of thought is described as mathematical realism, and can be defined as the belief that our mathematical theories are describing at least some part of the real world (http://web.calstatela.edu/faculty/mbalagu/papers/Realism%20and%20Anti-Realism%20in%20Mathematics.pdf pg. 36). There are several subdivisions among this group and more detail is given to this later. The second statement, that humans create math, is characteristic of mathematical anti-realism. By this mode of thought, math does not necessarily have any connection to the real world; it exists because we create it and it is true because we have made it to be true.

The Realists

The realists should probably be subdivided into two main groups: Platonists and everyone else, with the “everyone else” being a minority, so we should probably have a definition for mathematical Platonism. According to both Stanford’s and the internet’s encyclopedias of philosophy, mathematical Platonism is based on the following theses: Existence, Abstractness, and Independence. Basically, mathematical objects exist they are also abstract, and your language, thoughts, religion, or anything else doesn’t change what they are.  I should probably also mention that there are also subcategories amongst the Platonists, like traditional Platonism, full blooded Platonism, and some others, but I don’t want to get into that.  There are, however, mathematical realists who do not subscribe to Platonism. One such group is the physicalists. A strong proponent of physicalism was John Stuart Mill. The argument for this is that math is the study of ordinary physical objects and is therefore an empirical science. According to this mathematics is basically meant to discover laws that apply to all physical objects. For instance, 1+1=2 gives us the law of all physical objects that when you have 1 of the object and you and another of the object you have 2 of the object instead. This differs from Platonism in that these objects are no longer abstract, but rather describe all objects. These are not the only two categories of realists. The main problem I have with this is that it means if all objects were to vanish math would cease to be true. This is because physicalism is not based on the abstractness of mathematical objects which means that the objects themselves must exist.

The Anti-realists

Anti-realism is in general the belief that Math does not have an ontology. As with mathematical realism there are a lot of different subcategories of mathematical anti-realism. I’ve chosen to talk a bit more about conventionalism and fictionalism because they seemed interesting.

Conventionalism holds that mathematical statements are true only because of the very definitions of the statements. By this mode of thought, math does not necessarily have any connection to the real world; it exists because we create it and it is true because we have made it to be true. The statement “pi is the ratio between the circumference of a circle and its diameter” is true only because we define a circle as being a shape with a radius r, a diameter 2r, and a circumference 2*pi*r, and not because the universe made it so. In this sense, the above statement makes about as much sense as “all bachelors are unmarried”; both are obviously true, however this is because of their definitions rather than being the result of some universal laws.

Fictionalism argues that statements like 1+1=2 make about as much sense as “Harry Potter’s owl was Hedwig”. Yeah it’s true, but only within its given context. It is important to note that statements such as 1+1=3 make about as much sense as “You’re NOT a wizard, Harry”, because given the context of the story, or fiction, these statements simply make no sense. There are some interesting similarities about Fictionalism and Platonism. The biggest one is that both of them take mathematical statements at face value. This is to say that both of them take 1+1=2 to mean that to add the mathematical object 1 and adding it to another mathematical object 1 will result in the mathematical object 2. The difference then is that where Platonism takes this to also mean that these abstract objects exist, Fictionalism does not accept that these objects exist. This is different from conventionalism in that conventionalism doesn’t even accept that you are referring to objects, regardless of their existence. The thing about Fictionalism is that the subject doesn’t technically actually even a little exist. By this I mean that Harry Potter doesn’t actually exist (probably) and that therefore he isn’t actually a wizard (probably) and that since he doesn’t exist he doesn’t actually own an owl named Hedwig (probably), and that by that same logic 1 doesn’t actually exist, and neither does 2, and 1+1 doesn’t equal 2 because none of them exist.

Implications of these schools of thought

Mathematical realism, in a certain sense, seeks to prove truths about the universe. This is most obvious when you consider modes of thought like physicalism, under which math would be a really general science, but even under Platonism you are seeking to find laws that govern these abstract objects you are finding. So for instance, when you have one of some object, and you add another of that object to that first object, you now have two of that object and according to mathematical realists, this is true. It is a fact. According to mathematical anti-realists, if you remove the humans, or whatever it is that is observing this addition, then there is no longer a group, one of the things, or two of the things. These concepts existed only because the humans said they existed, and when the humans stopped existing and thus stopped observing this these things lost the properties of being one, being grouped, and finally being two. The exact way in which this is argued depends on what subcategory one subscribes to. (https://www.youtube.com/watch?v=TbNymweHW4E&list=UU3LqW4ijMoENQ2Wv17ZrFJA)

How this relates to faith

Regardless of whether you believe that the statement “pi is the ratio between the circumference of a circle and its diameter” is true because of universal laws or because of human created definitions, the statement is still true. The importance of this is that it means that there, at least at this point in time, is no way to verify whether the reason for math existing is tied to the very nature of the universe or whether it is simply the product of the human mind. As a result of this, the belief in either of these theories is, at least in a certain sense, a leap of faith.

My thoughts on this

My personal opinion on this leans towards mathematical realism and more specifically Platonism. I agree that mathematical objects exist, but that they do not by necessity have a real world counterpart and thus are abstract, and I believe that regardless of whether or not humans exist, the mathematical concepts we have found to be true will still be true, even if no-one is around to appreciate, understand, or use them. One big reason I have for thinking this way is because of how various isolated cultures ended up discovering the same mathematical principles. By this I mean that counting systems, simplistic though they may have been, were not a unique event to just one area, but rather a common feature. I mean the Mayans had a counting system, so did the Greeks, Egyptians, Babylonians, Indians, etc. It seems somewhat unlikely to me that all these isolated cultures would create a method for defining something that doesn’t exist.

Additional reading/sources

Idea channel’s episode titled “Is Math a Feature of the Universe or a Feature of Human Creation?”

https://www.youtube.com/watch?v=TbNymweHW4E&list=UU3LqW4ijMoENQ2Wv17ZrFJA

Mark Balaguer’s “Realism and Anti-Realism in Mathematics”

http://web.calstatela.edu/faculty/mbalagu/papers/Realism%20and%20Anti-Realism%20in%20Mathematics.pdf

Stanford’s Encyclopedia of philosophy entry on Platonism and mathematics

http://plato.stanford.edu/entries/platonism-mathematics/

The Internet’s Encyclopedia of philosophy entry on Platonism and mathematics

http://www.iep.utm.edu/mathplat/#H1

Wikipedia’s entry on philosophy of mathematics. No, this was not used as a source; it is however, useful for additional reading.

http://en.wikipedia.org/wiki/Philosophy_of_mathematics

Mayans count as well

http://maya.nmai.si.edu/maya-sun/maya-math-game

Greeks count as well

A History of Mathematics, Merzbach and Boyer, pages 52-55

For Babylonian counting see

Plimpton 322

For Indians having a number system click the bbc  thing below

For the link to the bbc story thing

http://www.bbc.co.uk/programmes/b03c2zvr

for the comic

http://xkcd.com/900/