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If you are good at creating magical patterns with compasses and rulers, you will find interesting shapes with curved edges. Let's take a look at the simple geometry of arcs.

A simple example is a curved triangle (Reuleauxtriangle) consisting of the vertices of an equilateral triangle (shown in figure 1). Place the vertices of the compass on three vertices in turn, and then draw an arc connecting the remaining two vertices, resulting in a curved triangle composed of three arcs. The circumferential angles of the three arcs are all 60 °. A curved triangle is the only curved shape whose edge is made up of an arc and whose center is a vertex.

Figure 1: a curved triangle is made up of equilateral triangles. In general, the shape of an arc is mainly defined by two pieces of information: the radius of the circle and the arc angle. We can find other arc edge shapes that have an interesting relationship between the center of the arc and the vertices of the graph, such as the three shapes in figure 2. Draw a new arc from the vertex of the graph, and there is a rotation or mirror relationship between the resulting figure and the original shape. In addition, the center of each arc in the original shape is also located at the vertex of the new shape, so it is not difficult to see that the drawing process is reversible.

Figure 2: there are three shapes of "phantom". The newly drawn figure is called the "phantom" of the original figure, which is composed of arcs drawn by the vertices of the original figure, which is consistent with the shape of the original shape, but has the relationship of rotation or mirror image. The curved triangle mentioned above is self-referential: it is an illusion of itself. You may think that these phantoms exist illusory with the original shape, like a ghostly existence.

When mathematicians encounter two-dimensional graphics, they often ask themselves a question: similar to square tiles can cover the whole square. If you are given a tile of this shape, can you use it to lay out the whole plane?

For the four shapes mentioned so far, the answer is "no". None of these shapes can tile a complete plane alone. Covering a plane with arc-edge graphics requires the same amount of concave and convex arcs.

Next, let's look at the fifth shape, or the first type of shape.

A three-curve lens is the simplest geometric shape, which is made up of two identical arcs. Assuming that the radius of the arc is 1, the lens can be simply described by the angle of the arc. Take a point on the arc, divide the arc into two parts, and then each part can make a lens, subtract two small lenses from the large lens, and you will get an unusual three-curve shape.

Figure 3: three-curve geometry any three curves can be tiled periodically: if you tile the three-curve figure in a certain direction, you will get the shape shown in figure 4.

Figure 4: periodic tiling if the angle of the radians that make up the three curves is a factor of 360 degrees, and the arcs meet a special proportion (such as 1:2:3), then the curves will have the tiling properties of radial and aperiodic properties, which is very interesting. You can make this puzzle and try it.

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2 figure 5: each curve of the puzzle can be described by three arc angles in ascending order, in which the sum of two concave arcs is the value of a large convex arc. Most of the puzzles made so far are made of three curves of 30 °- 60 °- 90 °(Fig. 51) or 36 °- 72 °- 108 °Radians (Fig. 52). Of course, other angles or proportions can also be used. In addition to single-sided tiling, you can also tile in many types of ways.

What does the illusion of the three curves look like? Each curve has a corresponding phantom, which is rotated 180 °according to a fixed center of rotation to get the phantom. According to the difference of the curve, the phantom may be separated from the original shape or may overlap.

Figure 6: some three-curve phantom in order to determine or visualize the illusion of the three curves, only need to rotate the three curves half a circle, and then position the vertex connecting the two concave surfaces in the center of the original large arc. For example, for any triple curve with a 180 °large arc, the middle vertex of the phantom is located at the center of the semicircle, regardless of the two smaller arcs.

Figure 7: some three curves with 180 °large arcs and phantom for some symmetrical three curves, the phantom is what the original shape looks like after 180 °rotation, and it is also the mirror symmetry pattern of the initial image along the axis of symmetry of the original three curves.

Figure 8: some symmetrical trilinear curves and their phantom. Tile with Phantom when using three curves to tile a plane, observe the phantom of the tile three curves, and you will find that the phantom is also tiling. As predicted, cycle tiling also causes phantom cycles to line up.

Figure 9: periodic tiling with phantom, but if you take a closer look at this shape, you will find that things are not that simple. In figure 9, you have to look carefully to see which phantom matches the original shape. You will find that in addition to the whole set of original three curves rotated 180 °, the phantom position has also changed. If you reverse rotate the phantom position 180 °, you will find that the arrangement position has changed.

If you use three curves of different sizes, this phenomenon will be more obvious. In figure 10, you can see that the phantom is not only rotated 180 °, but also on opposite sides of each other, no longer sharing an arc.

Figure 10: the strange behavior of the transposed phantom also appears in the radial tiling. For figure 11, the illusion of a three-curve in the shape of a star or petal is a ring.

Figure 11: the pattern triple curve with circular phantom. Therefore, when the triple curve is used for tiling, the corresponding phantom is also tiled in a non-intuitive and transposed way. When you think about it, when you tile a material in the shape of a three-curve, the corresponding phantom is tiled in a strange way, which is creepy.

What happens when a particular shape is filled with a triple curve?

Fill circle if there is an object in the shape of a lens, two smaller lenses can be obtained by filling it with a three-curve curve. Each small lens can be filled with two smaller lenses and a curve. By analogy, any lens can be filled with a series of smaller and smaller lenses. This also applies to circles, because circles are also a kind of lens.

Figure 12: there are many ways to fill circles in the layers of lenses, and a common method is introduced here. First fill it with the largest triple curve, then look down for the largest arc at the next level to fill it, and so on, leave the "remaining lens" around the lower half. The remaining lenses are then filled with smaller curves until they become an infinite sequence.

Here are four methods, each of which is introduced only to a certain degree of filling.

Scheme An is filled by symmetrical method, and the arc of each layer is divided into two, as shown in the following figure: the arc angle of the first three-curve is 90 °- 90 °- 180 °, the second layer is 45 °- 45 °- 90 °, and the third layer is 22.5 °- 22.5 °- 45 °.

Figure 13: a: symmetrical filling of circles or, you can keep the minimum arc angle of the three curves at 22.5 °, using the arc starting from the same point (left of figure 14), or the form of large three curves interlaced (right of figure 14).

Figure XIV: options B and C. In both cases, seven three curves were used, but the arrangement was different. In both cases, the top three curves were 22.5 °- 157.5 °- 180 °. At the same time, in the above three filling methods, the number and size of the unfilled lenses in the second half of the week are the same, just eight 22.5 °lenses, and these lenses will eventually be filled with infinitely small three curves.

Scheme D is a variation of scheme B, but it is filled with a finer three-curve. At first, we can fill it with a small angle of 5 °(a small arc of 5 °), and the largest three curves are 5 °- 175 °- 180 °.

Figure 15: the thin three-curve chart 15 does not show the arcs and lenses in the lower half of the circle, because they are made very small, so it is difficult to distinguish at this scale. As the angle of the small arc approaches to 0 °, the main three curves will approach to infinity, and the remaining lenses to be filled will be infinitely small. This seems to be the simplest and most elegant way to fill a circle.

At this point, what does the phantom that fills the three curves look like?

Consider the phantom in the three cases of Amurc above, as shown in figure 16.

Figure 16: the symmetrical three-curve filling in Phantom An of scheme An is relatively simple. Each phantom is the mirror symmetry of the original shape, and the vertices are on the symmetry axis of the original shape. The phantom of options B and C is shown in figure XVII.

Figure 17: the phantom of B and C can be found that the phantom generated in all three cases is the same. why?

Figure 18: what happens if the mysterious outline of the phantom continues to fill the remaining lenses of the figure? Taking scenario An as an example, the drawing that continues to be filled is shown in figure 19.

Figure XIX: filling the remaining "cracks" will get the same result for Amurc to continue filling. For each filling method, the Phantom fills the gap between the bulges on the outside of the new shape, or extends the tip upward. The limit shape is a semicircle with a radius of the diameter of the original circle, minus two original semicircle shapes, as shown in figure 20. Whether the circle is filled with symmetrical or asymmetric tri-curves, the phantom results are the same. As a result, the Phantom creates and fills a new shape: a symmetrical square (arbelos, a shape made up of three semicircles).

Figure 20: the resulting super phantom when scheme D uses the same limit case, the result is the same: as the number of filled curves is close to infinity, the shape of the phantom is close to the symmetrical square circle.

Figure 21: we can do the conversion of scheme D the other way around, though it will be a little difficult: we can start with a symmetrical square circle, fill it with infinite curves, then make the phantom of these curves, and finally go back to the filling circle. This is true no matter how the circle is filled with a three-curve. Because the symmetrical square circle is the union of all the phantom curves in the filling circle, we call the symmetrical square circle the super phantom of the circle. And vice versa, because it is reversible. Therefore, the super phantom of a shape is the outline of the phantom that fills the three curves of the shape.

Go back to the lens and apply the above operation on the circle to the lens as well. Going back to scenario B, find a lens (orange outline) from the entire circle, as shown in the corresponding super phantom figure 22.

Figure 22: sub-lenses and their super phantom more sub-graded lenses and their phantom.

Figure 23: if you compare another level of lens and its super phantom side by side, you will find the result of figure 24.

Figure 24: there are many ways to fill different lenses and phantom lenses with hyperbola under scheme B. We can also fill lenses with lenses less than 22.5°. As we do when we fill a circle, we can fill it to the limit. Finally, for any lens, the super phantom is a square circle of generalized symmetry, in which the Radian may be less than 180 °. There is an initial lens made up of three arcs with the same angle, and the relationship between the lens and its phantom is shown in figure 25.

Figure 25: lenses and their super phantom minus super phantom finally, we notice that lenses and super phantom can be subtracted from larger lenses and super phantom. In the following figure, two smaller lenses (center) are removed from the original lens (left), resulting in a curve (right), while the super phantom changes accordingly.

Figure 26: remove two lenses and look at another example, subtract two lenses from one circle, and the corresponding super phantom subtracts two generalized symmetrical squares from a completely symmetrical square.

Figure 27: subtracting two lenses from the circle you may notice that in the two images above, two smaller lenses are removed from the larger lenses, and the rest form a curve. The resulting super phantom is also the illusion of the three curves. It turns out that for any three curves, the super phantom is the same as the phantom!

This article starts with simple rulers and compasses, from arcs to lenses, to curves, to phantom, to filling circles, super phantom to square circles. But you can go one step further and combine the circle with its super phantom. Give it a try.

Author: Tim Lexen

Translation: Nuor

Revision: ClearC

Original text link:

Https://plus.maths.org/content/ghosts-tileshttps://plus.maths.org/content/ghosts-tiles-continued

This article comes from the official account of Wechat: Institute of Physics, Chinese Academy of Sciences (ID:cas-iop), author: Tim Lexen

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