Posts tagged: math
There will be an annular solar eclipse taking place this evening (5/20/2012) viewable from the western parts of the US and eastern Asia.
This is what happens when the Moon blocks some of the light from the Sun by passing in front of it from our vantage point on Earth. It is an annular eclipse and not a total eclipse because the Moon won’t completely cover up the Sun in the sky as it does during a total solar eclipse. Solar eclipses happen in these two varieties because the distance from the Earth to the Moon varies during its orbit around Earth. Therefore, the apparent size of the Moon looks different depending on how close the Moon is to Earth, and will only cover up as much of the Sun as it can. In fact, the Moon happens to be near its furthest point now.
It is interesting to note how close the Moon comes to completely covering up the Sun, and how this depends on the geometry of the situation.
During a total solar eclipse, the Moon will cover up the Sun almost exactly without much overlap. This happens because the apparent size of the Moon as viewed from Earth is nearly the same as the apparent size of the Sun as viewed from Earth.
This can be measured by comparing the ratio of the Sun’s radius and its distance to Earth to the ratio of the Moon’s radius and its distance to Earth. According to my own calculations they only differ by factor of about 3% of the Moon ratio. This is justified since the the Moon happens to be about 400 times smaller than the Sun, but is also about 400 times closer. Bigger differences in these ratios would imply that the Moon looks smaller or larger than the Sun during an eclipse. This closeness is equivalent to the claim that the two right triangles drawn in this diagram are close to being similar.
Is there any physical reason for why these ratios are the way they are? It seems plausible that the Moon could have had a different size, and orbited a little closer or further away from Earth thereby preventing such a ‘perfect’ total eclipse from happening. The configurations we witness almost seem like some coincidence!
So what? Is there any value to this special orientation during an eclipse?
Actually, the perfect total eclipses we are lucky enough to experience are valuable opportunities for astronomical observation.
In 1919, Arthur Eddington observed a total solar eclipse and was able to experimentally verify the phenomenon of gravitational lensing—one of the theoretical predictions of general relativity, which involved the bending of distant star light due to the Sun’s gravity. In addition, these perfect total eclipses also allow for other observations of solar phenomenon.
If it were not for this ecliptic coincidence and things were any different, then how much more difficult would it have been for scientists to learn about these other astrophysical phenomenon?
Take some region of the plane, and any number of distinct lines that pass anywhere through this region. Consider these random lines for instance:
Notice how the lines and their crossings create polygonal shapes in the region.
Using just two colors, say black and white, is it possible to color the entire region such that any two polygons that are next to each other sharing a common edge are different colors?
This is possible in the above example as this 2-coloring indicates:
There is also another possible 2-coloring that satisfies these requirements, but its really just the same as the 2-coloring above with the colors switched.
You can convince yourself that these are the only two permissible 2-colorings meeting the criterion with this particular configuration of lines.
One may wonder if its always possible to achieve such a 2-coloring, or precisely under what circumstances it is or is not possible.
It does seem to be the case that if each line passes completely through the region, then a 2-coloring will always be possible.
The following examples show this for two particular cases, where not only are there a different number of randomly chosen lines in each case, but each line is even allowed to move. Regardless, at each instance, the 2-coloring is always preserved!
Still, these examples do not prove the claim in general since there remains an infinite number of cases left unconsidered. How would one prove this?
Well, when would such a 2-coloring not be possible?
At any intersection of lines in the region, the crossings create corners for the polygons that are formed. If there happens to be an intersection with an odd number of corners, then for any assignment of 2 colors to the parts around this intersection, there would have to exist two adjacent parts that have the same color. Otherwise, for an even number of corners around an intersection, it is always possible to assign a 2-coloring so that adjacent parts have different colors.
Therefore, as long as all the intersections formed within the region have an even number of corners, there will exist a 2-coloring. This criterion will be met if we assume that the lines always pass completely through the region as in previous considerations
These conditions are special and do limit the possible configurations that are 2-colorable.
What if configurations were allowed to have intersections with an odd number of corners?
What if lines didn’t have to pass completely through the region and were allowed to end somewhere inside of it?
What if we didn’t have to use straight lines to partition the region?
If it is not possible to color the region in the above sense with 2 colors, how many would it take?
Does there exist some maximum finite number of colors that can be used to color any possible partition of a region?
The 4-color theorem, first stated in 1852, which concerns the problem under consideration, states that only 4 colors are needed to color any configuration so that adjacent regions are not colored the same.
The truth of this theorem went without correct proof until 1976 when it was proved by Kenneth Appel and Wolfgang Haken using a computer! This computer-assisted proof may be considered controversial and has interesting implications.
It is worth mentioning that the related problem of deciding whether a given configuration is 2-colorable is easy to solve since there are efficient computer algorithms that can check. However, the problem of deciding if 3 colors are needed is hard to do in general since there are currently no known computers algorithms that can efficiently solve this problem.
If you can find a fast algorithm, or if it you can prove that no efficient algorithm can exist for deciding the 3-coloring problem, then you could win $1,000,000 solving a big open problem in computer science.
![In hopes of making this blog more well rounded for the likes of Tumblr, graphed above is a plot resembling the shape of a cannabis leaf.
It is given by the following equation in polar form:
or as a parametric curve in Cartesian coordinates as
Open problems: rigorously defining pizza and cats using mathematics.
Mathematica code:
ListAnimate[ Table[ PolarPlot[ (1 + .9 Cos[8 *t]) (1 + .1 Cos[24 t]) (.9 + .05*Cos[200 t]) (1 + Sin[t]), {t, 3 Pi/2 - T, 3 Pi/2 + T}, PlotRange -> {{-2.5, 2.5}, {-.5, 4}}, PlotStyle -> {Green}, AxesStyle -> Directive[White], Background -> Black, ImageSize -> 500], {T, Pi/60, Pi, Pi/60}]]](http://25.media.tumblr.com/tumblr_m2stao5Sdf1qfjvexo1_500.gif)
In hopes of making this blog more well rounded for the likes of Tumblr, graphed above is a plot resembling the shape of a cannabis leaf.
It is given by the following equation in polar form:
or as a parametric curve in Cartesian coordinates as
Open problems: rigorously defining pizza and cats using mathematics.
Mathematica code:
ListAnimate[
Table[
PolarPlot[
(1 + .9 Cos[8 *t]) (1 + .1 Cos[24 t]) (.9 + .05*Cos[200 t]) (1 + Sin[t]),
{t, 3 Pi/2 - T, 3 Pi/2 + T},
PlotRange -> {{-2.5, 2.5}, {-.5, 4}}, PlotStyle -> {Green},
AxesStyle -> Directive[White], Background -> Black, ImageSize -> 500],
{T, Pi/60, Pi, Pi/60}]]
This will be an attempt at describing the algorithmic procedure used to generate some of the graphics posted here.
- Start with a line segment of any length L.
- Now pick any angle A in the range 0° to 180°.
- Connect another line segment of the same length as the first at the end of the first line so that two create an angle of A between them.
- Repeat this procedure connecting additional lines of length L at the same angle A to the endpoints of the previous lines.
This shows the procedure repeated several times using an angle of about 20°.
And this shows the pattern that results some 150 lines into the procedure with the same angle of 20°:
Instead, if each additional line segment were to be made slightly smaller than the previous, say 99% of the length of the previous line, then the lines would look something like this where they begin to spiral in towards the center:
Its interesting to observe how these patterns change as the value for the angle A is varied. The closer the values for two different angles are the closer the two patterns will resemble one another. However, for each distinct angle A the resulting pattern is unique.
Here are some animations that show the angle vary through some range while keeping the number of lines in the iteration fixed. Note how relatively small the range is that the angle varies through.
With all lines the same length varying through angles of about 10.7° to 10.4° :
and with each line 99% the length of the previous with an angle variation from 20° to 16° :
This algorithm can be equivalently thought of as taking a certain ordered sequence of points in the plane and then joining them with straight lines—playing connect the dots basically. Different arrangements and sequences of points in the plane would produce different patterns when connected with lines.
Perhaps the most elegant and concise way to mathematically describe this algorithm is by making use of the complex numbers. Due to the way complex numbers multiply, this algorithm can be specified by picking a complex number z and then successively multiplying it by itself to get a sequence of points given by zn, where the resulting complex numbers zn represent the nth point in the sequence. Then the connect the dots routine is performed with this sequence of points.
For instance, the configuration for the first 100 dots corresponding to a particular choice of complex number may look something like this:
Then once all the dots are connected with lines it would look like this:
Performing this procedure with each different point in the complex plane generates a different pattern.
This kind of procedure where a certain transformation is repeated on some elements is considered an iterated function system, which is a class of fractals.
You could imagine all the different possibilities which would result from using different functions.
In fact, all of these images here were generated using similar procedures.
Download an interactive CDF file here where you can control the parameters, view the graphics, and also create numbered dot sequences.
For any orthogonal maze shape you can possibly think of, there exists a crease pattern on a flat piece of paper which can be folded into that maze shape. There also exists an algorithm generating the crease pattern given the maze shape that is easy to implement in terms of its algorithmic complexity. You can design your own here.
Shown above is a crease pattern that folds into the “maze” on the left, which is a representation of the letters on the right. Red lines are mountain folds and blue lines are valley folds. I personally just like looking at the crease patterns.
Recommend books about mathematics. I am specifically interested in books for the layperson, but books at other levels are fine too. Perhaps you could mention the reading level (something like “for anyone”, “basic math background”, “advanced”).
?
I am assuming you would like book recommendations for the average lay person about mathematics in general, and not math textbooks one would read to learn the detailed workings of a certain branch of mathematics.
One of my all time favorite books is “Gödel, Escher, Bach: An Eternal Golden Braid” by Douglas Hofstadter. It is not exclusively about mathematics, but is more about logic and philosophical themes like self-reference and the emergence of cognition. It inspired a whole generation of computer scientists after its publication some 30 years ago, and probably continues to do so today.
I have also read A Mathematician’s Apology by G. H. Hardy, which isn’t so much about mathematics but rather about why one studies mathematics.
Unfortunately, I have very limited experience with books for the lay person. I think some followers of this blog may be able to give better recommendations than I can at this moment. If you are willing, feel free to do so here or here .
Whether or not you know about or care to use the mathematical coordinate systems described in the last post, it seems that our eyes and brains already make use of them when seeing and hallucinating. How this is the case will be explained here in some detail. Of course, our senses are intricately correlated with the dynamics of our brains in ways so complex that we can barely understand or even comprehend them at all, but there are still ways to reason and model (however crude they may be) the workings of such phenomenon.
Visual hallucinations are a universal human experience which occur in varying degrees in different circumstances—from when we rub or apply pressure to our eyes to more extreme instances like when we are under the influence of psychoactive drugs. What we “see” when we hallucinate is ultimately a unique subjective experience, but there does seem to be some objective similarities amongst reported hallucinations. Some of these hallucinations appear to be fixed in the visual field and do not change as we look around. Moreover, they can even be experienced when the eye are closed, and have also been reported by people who are blind. This gives reason to believe that the source or cause of these hallucinations may be independent of what we really see and receive as sensory input from the external world, and instead originate on a deeper level within the brain itself.
Studies suggest that a certain region of the brain, the visual cortex (also known as V1), plays a primary role in the processing of visual information. There is a correspondence from what we see in the visual field with our eyes to the neural activity in parts of V1. Spatial input of light and dark on the eyes is translated to “light/dark” regions of high/low neural activity in V1. Basically, images form certain patterns on the retinas of our eyes which are then converted to related patterns on the visual cortex in our brains leading to the perception of the image.
To describe more rigorously the nature of this image translation from the eyes to the brain, we can use mathematics and coordinate systems to define a coordinate mapping between the visual field to the V1 region of the brain. Such a transformation was modeled by J. D. Cowan and G. B, Ermentrout in their 1979 paper “A Mathematical theory of Visual Hallucinations”. A simplified version capturing the essence of their model is described in what fallows.
Interpreting the visual field as a two-dimensional plane, let’s use polar coordinates to label points in the visual field. The center of the visual field is taken as the origin of the coordinate system so that a point P in the visual field is described by a pair of numbers P = ( r , a ), where r is the distance of the point P to the center of the visual field and a is the relative angle the point makes with respect to some fixed axis.
Now we need a way a describe how points in the visual field correspond to points in the V1 region of the brain, but first we need a way to label points in the V1 region. To do this, we model the V1 region as another two-dimensional flat plane. This time, let’s use the normal Cartesian coordinate system to label the points in V1 as pairs of numbers ( x , y ).
The coordinate mapping constructed here will tell us how a point in the visual field, labelled as P = ( r , a ) in polar coordinates, gets translated to the corresponding point in the V1 region of the brain labelled as ( x , y ) in Cartesian coordinates. This ( x , y ) point in V1 is given in terms of the r and a coordinates of the visual field through the relationship
( x , y ) = ( ln r , a ),
where ln r is the natural logarithm of r.
Thus, the point that is given by the polar coordinates ( r , a ) in the visual field is mapped to the corresponding point in V1 whose Cartesian coordinate has a horizontal component of ln r and a vertical component of a. This transformation can be interpreted as log-polar coordinates, and may be recognized as the complex logarithm. We can now use this coordinate transformation to describe the shapes of certain hallucinations.
Normally, we actually receive sensory input through the visual field and corresponding neural patterns are triggered in our brains which result in the perception of whatever we are seeing. The model explained here explains the existence of hallucinations by an opposite mechanism. Without even receiving real sensory input through the visual field, V1 can still be a brain region with high neural activity (perhaps more so when on certain drugs). These patterns of neural activity in V1 may be perceived as if one is actually seeing a pattern in the visual field, but they are really just hallucinations. In the 1920s, psychologist Heinrich Klüver researched himself and others while having ingested mescaline (more info here) in the form of peyote buttons and attempted to classify the visual hallucinations they experienced. The observed hallucinations manifesting themselves as geometrical patterns were classified into four types and were referred to as form constants: 1) tunnels, 2) spirals, 3) cobwebs, and 4) lattices.
The coordinate transformation just described between the visual field and the V1 region of the visual cortex is successful in explaining the occurrence of the form constants.
1) Vertical or horizontal stripes stripes of neural activity on the V1 region may look something like this:
Applying the coordinate transformation in the reverse direction, we see that vertical stripes get mapped to circles in the visual field and horizontal stripes get mapped to rays emanating out from the center of the visual field:
2) Stripes of neural activity in V1 in arbitrary directions such as these diagonals
get mapped to spirals in the visual field:
3) Combinations of stripes such as these vertical and horizontal ones
would then get mapped to patterns that resemble cobwebs:
4) Neural activity in V1 is not limited to stripes. There may be activity with certain lattice symmetries like these checkerboard or honeycomb patterns
which would get mapped to hallucinations that appear like these
This model and these examples are idealized cases, but serve to approximate what may be happening in our brains when certain hallucinations occur. Despite having empirical applications in this context, the coordinate transformation described here can be freely applied to any image resulting in transformed images that still manage to express a psychedelic aesthetic (see here for some examples).
If you ever wondered when or why you would use polar coordinates I hope this post serves as a justifiable application. Conversely, I do not necessarily justify the use of psychedelics to help understand what you are learning in your math lessons.
This post was inspired by these papers, which explain what was attempted here and more in much greater detail:
- “A Mathematical Theory of Visual Hallucinations”
- “Geometric Visual Hallucinations, Euclidean Symmetry and the Functional Architecture of Striate Cortex”
- “A Model for the Origin and Properties of Flicker-Induced Geometric Phosphenes”
Get the Mathematica code for these animations as a CDF file here.
For the purposes of describing something, sometimes its useful to have a systematic way to label the relevant parts of that thing. In mathematics, things such as curves, surfaces, and abstract spaces in general can be described using what are called coordinate systems.
A familiar example of such a coordinate system may be the standard way points are described in the two-dimensional real plane. This is illustrated in the animation below. Here, an arbitrary point in the plane P (in black) is described as a pair of numbers P = ( x , y ), where x (the point in red) and y (the point in blue) are real numbers which give the lengths of the horizontal and vertical components of the point P, respectively. Thus, any two numbers x and y uniquely determine some point P in the plane. This coordinate system is sometimes referred to as rectangular coordinates or Cartesian coordinates.
Like many things, there isn’t only one correct way to describe them. There may exist different descriptions that accomplish the same thing. The Cartesian coordinate system described above is not the only possible or worthwhile coordinate system.
Another coordinate system that can be used to describe points in the plane is called polar coordinates, which is shown below. In this system, any point in the plane P (in black) is specified by two numbers r and a. Here, r is the distance of the point P to the origin (the length of the red dashed line). The other number a gives the angle the red dashed line makes with respect to the positive horizontal axis, which is measured by the dashed arc in blue (ranging in value from 0 to 360 degrees). Hence, any point P can be specified by how far away it is from the origin and its relative orientation to the horizontal axis.
Although these coordinate systems go about describing the points in the plane differently, they are equivalent in the sense that they both describe the same space. In this case, it is possible to translate data in one coordinate system to the other. That is, given some point P = ( x , y ) in Cartesian coordinates, that same point P can be expressed in polar coordinates where the corresponding numbers r and a can be found in terms of just the x and y. It can be shown using some trigonometric identities and the Pythagorean theorem that this transformation is given by
\[ r = \sqrt{x^2 +y^2} \]
and
\[a =\arctan(y/x) *\]
Likewise, starting in polar coordinates with the two numbers r and a describing some point P, the corresponding Cartesian coordinates ( x , y ) for the same point P are given by
\[ x = r\cos(a) \]
and
\[ y = r\sin(a) \]
Which coordinate system in preferred is largely a matter of context or personal taste. For instance, polar coordinates are useful to describe things that exhibit some kind of circular symmetry because equations may be expressed more naturally and concisely than in other coordinate systems. The examples described here are just two of many other valid coordinate systems that are used. In three dimensions, some other common coordinate systems used to exploit certain symmetries are spherical and cylindrical coordinates.
* As pointed out by isolatedvertex, this equation is not entirely correct due to certain degenerate cases. See here for more of an explanation.
You can move points in the Cartesian and polar coordinate systems around yourself in this interactive document if you have the Wolfram CDF player installed on your computer.
Mathematica code for animations:
Cartesian coordinates;
Manipulate[
Graphics[{
Line[{{0, 0}, pt}],
PointSize[Medium], Point[pt], Style[ Text[P, pt + {.1, .1}], Bold, Larger],
{Red, Dashing[.02], Line[{pt, {Part[pt, 1], 0}}],
PointSize[Medium], Point[{Part[pt, 1], 0}],
Style[Text[x, {Part[pt, 1], 0} + {.1, .1}], Bold, Larger]},
{Blue, Dashing[.02], Line[{pt, {0, Part[pt, 2]}}],
PointSize[Medium], Point[{0, Part[pt, 2]}],
Style[Text[y, {0, Part[pt, 2]} + {.1, .1}], Bold, Larger]}},
Axes -> True, AxesStyle -> Gray, AxesLabel -> {OverHat[x], OverHat[y]},
PlotRange -> {{-2, 2}, {-2, 2}}],
{{pt, {1, 1}}, {-2, -2}, {2, 2}}]
Polar coordinates;
Manipulate[
Graphics[{
PointSize[Medium], Point[pt], Style[ Text[P, pt + {.1, .1}], Bold, Larger],
{Red, Dashing[.02], Line[{{0, 0}, pt}],
Style[Text[r, .5 pt + {-.1, .1}], Bold, Larger]},
{Blue, {Dashed,
Circle[{0, 0}, .4,
{0, If[Part[pt, 2] > 0, ArcTan[Part[pt, 1], Part[pt, 2]], 2 Pi + ArcTan[Part[pt, 1], Part[pt, 2]]]}]},
Style[Text[a,
{.5 Cos[.5*If[Part[pt, 2] > 0, ArcTan[Part[pt, 1], Part[pt, 2]],
2 Pi + ArcTan[Part[pt, 1], Part[pt, 2]]]],
.5 Sin[.5*If[Part[pt, 2] > 0, ArcTan[Part[pt, 1], Part[pt, 2]],
2 Pi + ArcTan[Part[pt, 1], Part[pt, 2]]]]}],
Bold, Larger]}},
Axes -> True, AxesStyle -> Gray, AxesLabel -> {OverHat[x], OverHat[y]},
PlotRange -> {{-2, 2}, {-2, 2}}],
{{pt, {1, 1}}, {-2, -2}, {2, 2}}]
(click through for hi-res images)
Examples of tilings constructed as projections of D-dimensional simple cubic lattices (for D = 3, 5, and 7) into d = 2-dimensional physical space.
From: Two-Dimensional Random Tilings of Large Codimension
Making use of music theory, group theory, and category theory
From Musical Actions of Dihedral Groups
Abstract:
The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.
Summary:
This paper connects the twelve musical tones to elements in the dihedral group of order 24 (the symmetries of a regular dodecagon). The translation from pitch classes to integers modulo 12 allows for the modeling of musical works using abstract algebra. The first action on major and minor chords described in the paper is based on the musical techniques of transposition and inversion. A transposition moves a sequence of pitches up or down and an inversion reflects a melody about a fixed axis. The other action arises from the P, L, and R operations of the 19th-century music theorist Hugo Riemann. It is through these operations that the dihedral group of order 24 acts on the set of major and minor triads. The paper also describes how the P, L, and R operations have beautiful geometric presentations in terms of graphs. In particular the authors describe a connection between the PLR-group and chord progressions in Beethoven’s 9th Symphony, which leads to a proof that the PLR-group is dihedral. Another musical example is Pachelbel’s Canon in D. In summary, the paper gives a very pretty explanation of what we commonly hear in tonal music in terms of elementary group theory.
From Sacred Geometry: Philosophy and Practice by Robert Lawlor
(via theuniverseworks)
The “I” is the root of one’s negativity. It is imaginary.
In math, the imaginary unit is defined as the square root of negative one:
\[ i = \sqrt{-1} \]
For some motivation, consider what values for x satisfy the following equation
\[ x^2 = 1 \]
Both +1 and -1 work.
Instead, try to think of values for x that satisfy this equation
\[ x^2 = -1 \]
No real number does! Squaring both positive and negative numbers always gives another positive number.
If we allow for the existence of some number with the property that squaring it gives -1, then a solution to the equation would be found. This is precisely what the imaginary unit is constructed to do, and is equivelent to the definition given above since
\( i^2 = -1 \) implies \( i = \sqrt{-1} \).
Adding this imaginary unit together with any real multiples of it to the usual set of real numbers gives you the complex numbers. This is a number system so rich in its structure that much of modern mathematics and science is dependent on its existence. For instance, the theory of quantum mechanics makes heavy use of imaginary numbers, many fractals are constructed using them, and even the pretty colors plotted on this site are exploiting them.
And they would say you couldn’t take the square root of negative numbers!
Thanks for the request! It is an interesting one, but I am not able to personally make an animation at the moment. Perhaps, I will in the future.
I immediately thought of this video when I read this question. It gives a visual proof of the fundamental theorem of algebra but seems to be more topological based using the concept of winding number to prove the claim. However, I must admit that the method of proof may not be so obvious without reading the video description before watching it.
Although all proofs of this theorem will make use of complex analysis more or less by definition, I think the proof given in this paper by Danial Velleman makes nice use of the complex numbers. He does this by assigning a unique color to each complex number, and then visualizing the image of the complex plane after evaluating it through the complex map defined by polynomial equations. This seems like a more intuitive approach than those taken in other proofs I have seen of this theorem.
I hope this at least temporarily satisfies whoever it is that cares. Is anyone else willing to share any neat visual proofs of the fundamental theorem of algebra?
