Posts tagged: math

nldmut:

Specular holograms by Matthew Brand currently on display at the new Museum of Mathematics in New York.

See his site for more.

The technique used by Brand to create these pieces is not one of conventional holography. He meticulously controls the unique shape of thousands of tiny optical pieces placed on a surface creating a 3D effect when the light source or viewer moves. This is essentially a mathematical problem in differential geometry and combinatorial optimization. Brand was the first person to correctly describe this technique in 2008 even though it dates back as early as the 1930s (check out his paper for details).

A Penrose tiling can be constructed using just two different tiles in the shape of a thick and thin rhombus:

Here, the angles are multiples of ø = π/5. The edges of these tiles are marked with two different kinds of arrows. This is done to enforce a specific matching rule that ensures that the tiling is non-periodic, which is one of the defining features of Penrose tilings. Different tiles can only be placed next to each other if the touching edges have the same type of arrow and point in the same direction.
If these matching conditions were not in place, then it would be possible to construct tilings which are periodic as shown in the following image. There are translational symmetries present in 8 different directions here:

The image at the top shows a part of a Penrose tiling obeying the matching conditions with the arrows displayed. The numbers are just there to index some other property of directionality not discussed here.
If these matching conditions are satisfied for a complete tiling, then the resulting configuration will always be non-periodic. However, following the matching rules alone does not guarantee an infinite tiling of the entire plane. It is therefore possible to construct finite regions that obey the matching rules, but cannot be extended any further without allowing for periodicities or contradicting the matching rules.
There do exist sets of tiles that will always admit non-periodic tilings in which no extra matching conditions need to be imposed. For a list of such tiles that tile the plane, 3-dimensional space, and even the hyperbolic plane, see this list of aperiodic sets of tiles.
Image sources:
Algebraic Theory of Penrose’s Non-periodic Tiligns of the Plane by N.G. de Bruijn
The Empire Problem in Penrose Tilings by Laura Effinger-Dean

A Penrose tiling can be constructed using just two different tiles in the shape of a thick and thin rhombus:

Here, the angles are multiples of ø = π/5. The edges of these tiles are marked with two different kinds of arrows. This is done to enforce a specific matching rule that ensures that the tiling is non-periodic, which is one of the defining features of Penrose tilings. Different tiles can only be placed next to each other if the touching edges have the same type of arrow and point in the same direction.

If these matching conditions were not in place, then it would be possible to construct tilings which are periodic as shown in the following image. There are translational symmetries present in 8 different directions here:

The image at the top shows a part of a Penrose tiling obeying the matching conditions with the arrows displayed. The numbers are just there to index some other property of directionality not discussed here.

If these matching conditions are satisfied for a complete tiling, then the resulting configuration will always be non-periodic. However, following the matching rules alone does not guarantee an infinite tiling of the entire plane. It is therefore possible to construct finite regions that obey the matching rules, but cannot be extended any further without allowing for periodicities or contradicting the matching rules.

There do exist sets of tiles that will always admit non-periodic tilings in which no extra matching conditions need to be imposed. For a list of such tiles that tile the plane, 3-dimensional space, and even the hyperbolic plane, see this list of aperiodic sets of tiles.

Image sources:

(click through the images to view in high-res)

Penrose tilings are an example of the non-periodic tilings discussed in the last post. Recall that these are tilings that cover the entire infinite plane leaving neither gaps nor overlaps. Whats nice about these tilings is that the set of tiles used to construct the Penrose tilings only consists of two different basic shapes consisting of quadrilaterals

Whats even more remarkable about Penrose tilings is that using just these two shapes it is possible to construct infinitely many different tilings that cover the infinite plane. This infinity is infinitely bigger in size than the countable infinity of the whole numbers ( 1, 2, 3, 4, and so on), but rather equivalent to the uncountable infinity associated to the real numbers (which includes all whole numbers, fractions, and decimals with infinitely many digits).

Despite the existence of these infinitely many different Penrose tilings, there is one peculiar property they all exhibit. Consider any finite region, or patch, of one particular Penrose tiling. Then it is possible to find an exact copy of this patch in any other different Penrose tiling! Moreover, this patch occurs infinitely often in different spots in any given tiling! Note that this is true of any patch of any size no matter how big as long as its finite. This implies that if you were only able to examine a finite part of any Penrose tiling, you could never really distinguish that entire tiling from any other tiling. Thus, different Penrose tilings are only perfectly distinguishable in the infinite limit of the entire plane.

The images shown above display finite regions of Penrose tilings. They are constructed using an elegant "cut-and-project" method (also used here), which involves projecting the points of the integer lattice in 5-dimensional Euclidean space, onto a certain 2-dimensional plane. Connecting adjacent projected points in this plane by lines then yields a Penrose tiling.

Further reading:

Image Source: Wikipedia

QuasiMusic is a Java applet that generates music, functioning a lot like a player-piano roll, by exploiting the patterns in quasiperiodic tilings of the 2 dimensional plane. The animations show two different instances of the applet in action.

These are tilings that fill the entire infinite plane using different shapes called tiles that are arranged in a certain way with out overlapping each other and without leaving any empty space.

If its possible to take a copy of a tiling and shift it over in some direction by a certain amount relative to the original tiling and have it line up exactly with the original, then that tiling is periodic. This minimum shifting distance needed to make the tiling match itself is called the period.

Whats special about quasiperiodic tilings is that there is no such period. So no matter which direction you shifted the tiling, it would never exactly line up with itself.

Using these quasiperiodic tilings has interesting consequences for the music they generate. Since the pattern never repeats, this implies that the music will never repeat. Although there may be moments which sound similar they can never be exactly the same when considered in a longer time interval.

To understand how the applet creates the music from a given tiling first assign to each type of tile a sound (instrument and pitch). Next, imagine a series of vertical lines passing over the tiling. Then as the whole tiling moves upwards passing a horizontal axis, play a sound each time one type of tiling changes to another type on one of these vertical lines. In the animations, the tiles light up when they trigger a sound, and the series of vertically arranged white dots specify when the tiles change. These white dots are reminiscent of the holes in a player-piano roll.

The applet allows the user to control many different parameters, and their are over 350 different instruments and sounds to choose from literally allowing for endless possibilities. Check it out here.

Also check out a recently released dark ambient album which makes use of QuasiMusic on some tracks by master mathematics expositor John Baez.

The end of this post concludes with a visualization of a certain kind of three dimensional standing wave produced by a point source. The rest of this post will explain some of the intuition behind the modelling.

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asks:
Is it possible to visualize 3 dimensional standing wave patterns with mathematica?

The end of this post concludes with a visualization of a certain kind of three dimensional standing wave produced by a point source. The rest of this post will explain some of the intuition behind the modelling.

A standing wave is what results when a traveling wave such as this

and its time-reversed analog traveling in the opposite direction

are considered together and allowed to interfere with each other constructively and de-constructively. The positive or ‘upward waving’ parts of one of the waves are cancelled out by the negative or ‘downward waving’ parts of other wave.

This is example of a 1 dimensional wave, the curve, which is ‘waving’ into a higher 2nd dimension represented by the vertical extent on the plane that the curve lies in.

In this way, a 2 dimensional wave can be visualized as a 2 dimensional surface which ‘waves’ into a higher 3rd dimension. Here is a certain kind of 2D traveling wave produced by a point source which emits symmetrical in 2 directions creating these circular shaped waves:

And a standing wave produced from the interference of an outward and inward moving wave:

Trying to continue this way in order to visualize a 3 dimensional wave results in some issues, because the 3 dimensional thing that is doing the ‘waving’ needs to wave in a higher 4th dimension which we can’t really visualize by simply drawing in the 4th dimension.

Thus far, we have been associating waviness to movement in a higher spatial dimension. Instead, assign a color to the thing that is doing the waving to represent its waviness. Doing this allows us to eliminate that higher spatial dimension needed in the previous wave visualizations, and alternatively associates this to a color dimension.

Using this method to visualize the 2 dimensional traveling wave resulting from a point source looks like this, where lighter regions can represent positive values and darker regions negative values:

Now we can visualize a 3 dimensional wave by assigning a 3 dimensional region colors according to the wave structure.

Below is a traveling wave resulting from a point source emitting symmetrically in 3 dimensions producing spherical wavefronts. One of the upper quadrants of the region has been removed in order to expose the internal structure of the wave, and the bounding box is present to allow for a better sense of depth.

The standing wave resulting from an outward and inward traveling wave would then look like this:

Increasing the size of the waving region so that it fills the bounding box looks a little more interesting. The faces of the cube show 2 dimensional projections of the 3 dimensional wave which are identical to the 2D analog.

A traveling wave:

And a standing wave:

What was exemplified here only considered a certain kind of wave, which is the most symmetric of waves. In general, a standing wave can be produced by taking any waveform and adding it to the waveform produced when the time direction is reversed.

Interactive Mathematica code: notebook, CDF

asks:
Do you think that mathematics as a form of abstract thought is opposit eto meditation?

I think its important to first acknowledge how broad, and hence ambiguous, the term “meditation” can be. Despite what one may naively think of as meditation, the act itself comes in many different varieties. In some cases different meditation practices may even seem contradictory to one another. Regardless, many do share common features in their essence and guiding principles. For the sake of being in a fair position to comment, and in order for me to remain true to myself I will answer in the context of Vipassana mediation, or what may be referred to more generally as insight/mindfullness meditation.

Vipassana is a form of meditation with the objective of self-purification through self-observation. It seeks to eradicate self-inflicted suffering through a sort of reverse conditioning process of the mind in order to appropriately deal with the sources of personal suffering which are cravings and aversions. The main characteristic mind set for practicing the technique is to develop equanimity, which can be thought of as this ideal neutral state between one’s cravings and aversions. I won’t introduce the practice anymore than this (the interested reader can check out this tumblr blog for an idea, or better yet try it out for yourself).

It is worth mentioning that even though Vipassana is often practiced as a sustained silent sitting meditation, this serves merely as a controlled setting in which one is able to deepen the practice in order to apply and cultivate the technique in everyday life.

I think the question of how mathematics as a form of abstract thought is opposite to meditation is dependent largely on how the individual engages in mathematical thought, and not really dependent on mathematics in its purity. This should apply to most things we do and can think about. That is, any opposition to meditation that may exist is a consequence of the individuals subjective relationship with the thing or thought and not necessarily in the thing or object of thought itself. 

It may seem kind of silly to regard mathematics as something that allows for cravings or aversions, but I think this is a prevalent phenomenon for both mathematicians and non-mathematicians dealing with mathematics. The latter notion of having aversions towards mathematics is obviously more common amongst those that do not like doing mathematics, but even I must admit at times to not wanting to calculate some integral using some kind of iterated integration-by-parts procedure with some trigonometric substitutions.  A more general  example could be something like test anxiety experienced by test takers and students. The issue of forming cravings is subtle and could be more complex, but most could at least attest that it feels good to solve a problem.

The act of performing mathematical thought, say, while practicing a sitting meditation where the meditator might not be trying to engage in mathematical thoughts could pose an obstacle. However this can apply for any arbitrary thought in a certain context. Suppose you are trying to count the number of combinatorial arrangements of something, or getting caught up trying to formally visualize Hopf vibrations of a 3-sphere and its stereographically projected counterpart in three dimensional Euclidean space. Now, imagine some time has  passed and realizing you are trying not to think about either of those things while attempting to meditate.

A more interesting direction for this question might be to ask about the ways mathematics as a form of abstract thought does not oppose mediation. Or maybe even in the ways it might supplement that kind of thing.

In short, doing mathematics can be opposed to meditation practices, but it doesn’t have to be. Conversely, meditating or exercising whatever kind of awareness constitutes meditating does not have to oppose pure mathematical thought.

Coincidental Solar Eclipse?

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?

Planar 2-colorability

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}]]

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}]]
Connecting the dots in the complex plane

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.

(Image source)

Math book recommendations

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”).

?

asks:
Can you recommend me some good books about mathematics? thanks!

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 .