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How to make impossible Wallpaper: story of forbidden symmetries
At first glance, to come up with the Wallpaper no more difficult than performing the tasks of kindergarten. Designers can choose any combination of colors and shapes for the initial bite and then just reproduce it in two ways. Depending on the pattern of the initial piece and the choice of destinations may appear, and additional symmetry, for example symmetry of the sixth order in the first picture or mirror on the second. Both pattern created by mathematician Frank Faris, University of California, Santa Clara.
Left – pattern Wallpaper with fold rotational symmetry of the sixth order around each of the brown-green rosettes. To the right is the pattern Wallpaper with mirror symmetry relative to the horizontal lines passing through each elliptical element of the ornament of stained glass.
Tiles Penrose demonstrate many examples of local symmetry of the fifth order, but they have not found the repetition of the pattern. When filling large areas in the plane the ratio of the wide tiles to the number of narrow approaches to the Golden section.
But, although it is possible to make a Wallpaper with the rotational symmetries of the second, third, fourth, or sixth orders of magnitude, it is impossible to create Wallpapers with the symmetry of the fifth order (the order indicates the number of times during a 360° rotation will happen the alignment of the pattern – approx. transl.). This limitation is known to mathematicians for almost 200 years as the "crystallographic restriction". The geometry of the Pentagon forbids patterns with the symmetry of the fifth order. The same is true for orders of seven or more.
However, the most interesting patterns, such as Penrose tiles, showing the local symmetry of the fifth order in many places and on varying scales, but without repeated patterns. Using different from the approach of Penrose's method, Farris curbed the unusual geometry of the symmetry of the fifth order and created a new set of exciting images – pseudo-Wallpapers, not subject, at first glance, the crystallographic restriction.
Fig. 4
4th picture looks like a counterexample to the crystallographic restriction, having a rotational symmetry of the fifth order around the point And, despite the fact that the pattern can be shifted in the plane in directions AB or AC. In fact, Faris writes in his article for the journal Notices of the American Mathematical Society that this picture is just a clever fake.
"You know that you're observing symmetry is impossible," says Stephen Kennedy, Carleton College in Minnesota.
The rotational symmetry of the fifth order around the point And seems to be running. But if you look closely, you can see that wheels around points b and C are slightly different from A. If we could move away from the pattern to see a larger number of repetitions, the visible repetition of the pattern would be less and less similar to the pattern in the area of point A, even if all the more compelling copy And would appear in other places, as in Fig. 5. Faris has shown that such illusions can be created on a larger scale, removed from the pattern and repeating it a certain number of times – specifically, the number of times corresponding to numbers from Fibonacci series (1, 1, 2, 3, 5, 8, 13, 21,... where each successive number is the sum of the previous two), which also plays a role in the geometry of Penrose tiles.
Fig. 5
"Mind we understand that this is some kind of a hoax," says Faris. However, as he writes in the article, these images are "invite our eye to their study and enjoyment of almost perfect repetition".
Faris came up with these fakes, the changing technology, with which he created these Wallpapers with a rotational symmetry of 3-order, such as in Fig. 6.
To create symmetry of 3-order Faris started work in three-dimensional space, which has one particularly natural rotation, using three spatial coordinates, and the rotating point in space 120 degrees around the diagonal. Then Faris has created three-dimensional Wallpaper patterns, imposing a special way of the selected sinusoid, and combining them with a pre-selected palette of colors. Points were colored according to their position in the superimposed sinusoids. Then Faris led flat Wallpaper by limiting the two-dimensional color plane perpendicularly intersecting the axis of rotation of the original space.
This smooth, use a sine wave approach to the creation of the patterns of the Wallpaper differs from the traditional method of copying and pasting, says Kennedy. "This is a very new way to create symmetrical patterns."
Fig. 6
The same procedure carried out in the five-dimensional space, like it was supposed to lead to the creation of a pattern with symmetry of the fifth order – if only we didn't know that this is impossible. Interesting, thought Faris, at what point the system fails?
In theory, five-dimensional space possible, although its hard to imagine. He has a natural analog of the symmetry of rotation of the fifth order, as in three – dimensional space- symmetry of the third. In the five-dimensional space you can choose one of two planes, each perpendicular to the rotation axis and the other plane. Each of them can be rotated around a point on 72 or 144 degrees. It may seem difficult to imagine two planes and a straight line, perpendicular to each other, but in five dimensions, they all have enough space.
Faris realized what the problem is – if perpendicular to the plane neatly cuts the three-dimensional space, and contains an infinite Wallpaper infinite number of points having integer coordinates, two perpendicular planes in five-dimensional space of the irrational, and do not contain points with integer coordinates (except starting point). Since pattern Wallpaper created from sine waves, using repeated shifts on integers, such plane shall not inherit patterns from different spaces of the highest order.
"And so there is a fly in the soup," writes Faris in the article.
However, these two planes appears the illusion of the structure of Wallpaper, thanks to the participation of the so-called Golden ratio, the irrational number that describes the direction of the two planes, and Fibonacci numbers.
Also interesting: the Fibonacci Numbers
The FIBONACCI SPIRAL is encrypted the law of nature
Build relationships with them, Faris was able to show that although the two planes and no points with integer coordinates, each of them is very close to the infinite scattering of points with integer coordinates, the coordinates of which are Fibonacci numbers. Every time the plane approaching one of these points, Fibonacci, pattern looks almost the same as in the reference point, which creates the illusion of an exact copy.
Faris also figured out how to combine the colors and patterns of nature photos with the wave functions to be included in the design of the patterns, whereby it is possible to obtain a huge amount of "fake" Wallpaper. In the above illustration you can see the tree branches moving with the pictures.published
Translation: Erica Klarreich
P. S. And remember, just changing your mind - together we change the world! ©
Source: geektimes.ru/post/284594/
Left – pattern Wallpaper with fold rotational symmetry of the sixth order around each of the brown-green rosettes. To the right is the pattern Wallpaper with mirror symmetry relative to the horizontal lines passing through each elliptical element of the ornament of stained glass.
Tiles Penrose demonstrate many examples of local symmetry of the fifth order, but they have not found the repetition of the pattern. When filling large areas in the plane the ratio of the wide tiles to the number of narrow approaches to the Golden section.
But, although it is possible to make a Wallpaper with the rotational symmetries of the second, third, fourth, or sixth orders of magnitude, it is impossible to create Wallpapers with the symmetry of the fifth order (the order indicates the number of times during a 360° rotation will happen the alignment of the pattern – approx. transl.). This limitation is known to mathematicians for almost 200 years as the "crystallographic restriction". The geometry of the Pentagon forbids patterns with the symmetry of the fifth order. The same is true for orders of seven or more.
However, the most interesting patterns, such as Penrose tiles, showing the local symmetry of the fifth order in many places and on varying scales, but without repeated patterns. Using different from the approach of Penrose's method, Farris curbed the unusual geometry of the symmetry of the fifth order and created a new set of exciting images – pseudo-Wallpapers, not subject, at first glance, the crystallographic restriction.
Fig. 4
4th picture looks like a counterexample to the crystallographic restriction, having a rotational symmetry of the fifth order around the point And, despite the fact that the pattern can be shifted in the plane in directions AB or AC. In fact, Faris writes in his article for the journal Notices of the American Mathematical Society that this picture is just a clever fake.
"You know that you're observing symmetry is impossible," says Stephen Kennedy, Carleton College in Minnesota.
The rotational symmetry of the fifth order around the point And seems to be running. But if you look closely, you can see that wheels around points b and C are slightly different from A. If we could move away from the pattern to see a larger number of repetitions, the visible repetition of the pattern would be less and less similar to the pattern in the area of point A, even if all the more compelling copy And would appear in other places, as in Fig. 5. Faris has shown that such illusions can be created on a larger scale, removed from the pattern and repeating it a certain number of times – specifically, the number of times corresponding to numbers from Fibonacci series (1, 1, 2, 3, 5, 8, 13, 21,... where each successive number is the sum of the previous two), which also plays a role in the geometry of Penrose tiles.
Fig. 5
"Mind we understand that this is some kind of a hoax," says Faris. However, as he writes in the article, these images are "invite our eye to their study and enjoyment of almost perfect repetition".
Faris came up with these fakes, the changing technology, with which he created these Wallpapers with a rotational symmetry of 3-order, such as in Fig. 6.
To create symmetry of 3-order Faris started work in three-dimensional space, which has one particularly natural rotation, using three spatial coordinates, and the rotating point in space 120 degrees around the diagonal. Then Faris has created three-dimensional Wallpaper patterns, imposing a special way of the selected sinusoid, and combining them with a pre-selected palette of colors. Points were colored according to their position in the superimposed sinusoids. Then Faris led flat Wallpaper by limiting the two-dimensional color plane perpendicularly intersecting the axis of rotation of the original space.
This smooth, use a sine wave approach to the creation of the patterns of the Wallpaper differs from the traditional method of copying and pasting, says Kennedy. "This is a very new way to create symmetrical patterns."
Fig. 6
The same procedure carried out in the five-dimensional space, like it was supposed to lead to the creation of a pattern with symmetry of the fifth order – if only we didn't know that this is impossible. Interesting, thought Faris, at what point the system fails?
In theory, five-dimensional space possible, although its hard to imagine. He has a natural analog of the symmetry of rotation of the fifth order, as in three – dimensional space- symmetry of the third. In the five-dimensional space you can choose one of two planes, each perpendicular to the rotation axis and the other plane. Each of them can be rotated around a point on 72 or 144 degrees. It may seem difficult to imagine two planes and a straight line, perpendicular to each other, but in five dimensions, they all have enough space.
Faris realized what the problem is – if perpendicular to the plane neatly cuts the three-dimensional space, and contains an infinite Wallpaper infinite number of points having integer coordinates, two perpendicular planes in five-dimensional space of the irrational, and do not contain points with integer coordinates (except starting point). Since pattern Wallpaper created from sine waves, using repeated shifts on integers, such plane shall not inherit patterns from different spaces of the highest order.
"And so there is a fly in the soup," writes Faris in the article.
However, these two planes appears the illusion of the structure of Wallpaper, thanks to the participation of the so-called Golden ratio, the irrational number that describes the direction of the two planes, and Fibonacci numbers.
Also interesting: the Fibonacci Numbers
The FIBONACCI SPIRAL is encrypted the law of nature
Build relationships with them, Faris was able to show that although the two planes and no points with integer coordinates, each of them is very close to the infinite scattering of points with integer coordinates, the coordinates of which are Fibonacci numbers. Every time the plane approaching one of these points, Fibonacci, pattern looks almost the same as in the reference point, which creates the illusion of an exact copy.
Faris also figured out how to combine the colors and patterns of nature photos with the wave functions to be included in the design of the patterns, whereby it is possible to obtain a huge amount of "fake" Wallpaper. In the above illustration you can see the tree branches moving with the pictures.published
Translation: Erica Klarreich
P. S. And remember, just changing your mind - together we change the world! ©
Source: geektimes.ru/post/284594/
