An Apollonian Seal is a type of fractal image, formed by circles that become smaller and smaller contained in a single large circle. Each circle in the Apollonian Seal is "tangent" to the adjacent circles - in other words, these circles touch each other in infinitely small points. Named Apollonian Seal in honor of the mathematician Apollonius of Perga, this type of fractal can be brought to a reasonable level of complexity (by hand or computer) and forms a wonderful and impressive image. Read Step 1 to get started.
Steps
Part 1 of 2: Understanding the Key Concepts
"To be clear: if you are simply interested in" designing "an Apollonian Seal, there is no need to search for the mathematical principles behind the fractal. However, in case you want to fully understand the Apollonian Seal, it is important that you understand the definition. of different concepts that we will use in the discussion ".
Step 1. Define the key terms
The following terms are used in the instructions below:
- Apollonian seal: one of several names that apply to a type of fractal composed of a series of circles nested within a large circle and tangent to each other. These are also called "Plate Circles" or "Kissing Circles".
- Radius of a circle: the distance between the center point of a circle and its circumference, which is usually assigned the variable "r".
- Curvature of a circle: the function, positive or negative, inverse to the radius, or ± 1 / r. Curvature is positive when calculating external curvature, negative when calculating internal.
- Tangent - a term applied to lines, planes, and shapes that intersect at an infinitesimal point. In the Apollonian Seals, this refers to the fact that each circle touches all neighboring circles at one point. Note that there are no intersections - tangent shapes do not overlap.
Step 2. Understand Descartes' Theorem
Descartes' theorem is a useful formula for calculating the size of the circles in the Apollonian Seal. If we define the curvatures (1 / r) of any three circles - respectively "a", "b" and "c" - the curvature of the circle tangent to all three (which we will call "d") is: d = a + b + c ± 2 (sqrt (a × b + b × c + c × a)).
For our purposes, we will generally only use the answer we will get by placing a '+' sign in front of the square root (in other words, … + 2 (sqrt (…)). For now it is enough to know that the form equation negative has its usefulness in other contexts
Part 2 of 2: Building the Apollonian Seal
"Apollonian Seals are shaped like magnificent fractal arrangements of circles that gradually shrink. Mathematically, Apollonian Seals are infinitely complex, but, whether using a drawing program or drawing by hand, you can get to a point where it will be. Impossible to draw smaller circles. The more precise the circles, the more you will be able to fill to seal ".
Step 1. Prepare your drawing tools, analog or digital
In the steps below, we will make a simple Apollonian Seal. It is possible to draw an Apollonian Seal by hand or on the computer. Either way, make an effort to draw perfect circles. It is quite important because each circle in the Apollonian Seal is perfectly tangent to the circles that are close to it; circles that are even slightly irregular can ruin your final product.
- If you are drawing on a computer, you will need a program that allows you to easily draw circles with a fixed radius from the center point. You can use Gfig, a vector drawing extension for GIMP, a free image editing program, as well as a host of other drawing programs (see the materials section for some helpful links). You'll probably also need a calculator and something to write down radii and curvatures.
- To draw the Seal by hand you will need a scientific calculator, a pencil, a compass, a ruler (preferably with a millimeter scale), paper and a notepad.
Step 2. Start with a large circle
The first task is easy - just draw a large circle that is perfectly round. The larger the circle, the more complex the seal will be, so try to draw a circle as large as the page you are drawing on.
Step 3. Draw a smaller circle inside the original one, tangent to one side
Then draw another circle inside the smaller one. The size of the second circle is up to you - there is no exact size. However, for our purposes, let's draw the second circle so that its center point is halfway through the radius of the larger circle.
Remember that in Apollonian Seals, all touching circles are tangent to each other. If you are using a compass to draw your circles by hand, recreate this effect by placing the tip of the compass in the middle of the radius of the larger outer circle, then adjusting the pencil so that it just "touches" the edge of the large circle and finally, drawing the smallest circle
Step 4. Draw an identical circle that crosses the smaller circle inside
Next, we draw another circle that crosses the first one. This circle should be tangent to both the outermost and innermost circles; this means that the two inner circles will touch exactly in the middle of the larger one.
Step 5. Apply Descartes' Theorem to find out the dimensions of the next circles
Stop drawing for a moment. Remember that Descartes' Theorem is d = a + b + c ± 2 (sqrt (a × b + b × c + c × a)), where a, b and c are the curvatures of your three tangent circles. Therefore, to find the radius of the next circle, we first find the curvature of each of the three circles we have already drawn so that we can find the curvature of the next circle, then convert it and find the radius.
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We define the radius of the outermost circle as
Step 1.. Since the other circles are inside the latter, we are dealing with its "internal" (rather than external) curvature, and as a result, we know that its curvature is negative. - 1 / r = -1/1 = -1. The curvature of the large circle is - 1.
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The radii of the smaller circles are half as long as the large one, or, in other words, 1/2. Since these circles touch the larger circle and touch each other, we are dealing with their "outer" curvature, so the curvatures are positive. 1 / (1/2) = 2. The curvatures of the smaller circles are both
Step 2..
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Now, we know that a = -1, b = 2, and c = 2 according to the equation of Descartes' Theorem. We solve d:
- d = a + b + c ± 2 (sqrt (a × b + b × c + c × a))
- d = -1 + 2 + 2 ± 2 (sqrt (-1 × 2 + 2 × 2 + 2 × -1))
- d = -1 + 2 + 2 ± 2 (sqrt (-2 + 4 + -2))
- d = -1 + 2 + 2 ± 0
- d = -1 + 2 + 2
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d = 3. The curvature of the next circle will be
Step 3.. Since 3 = 1 / r, the radius of the next circle is 1/3.
Create an Apollonian Gasket Step 8 Step 6. Create the next set of circles
Use the radius value you just found to draw the next two circles. Remember that these will be tangent to the circles whose curvatures a, b and c were used for Descartes' Theorem. In other words, they will be tangent to the original circles and the second circles. To make these circles tangent to the other three, you will need to draw them in the blanks of the larger circle area.
Remember that the radii of these circles will be equal to 1/3. Measure 1/3 on the edge of the outermost circle, then draw the new circle. It should be tangent to the other three circles
Create an Apollonian Gasket Step 9 Step 7. Continue adding circles like this
Because they are fractals, the Apollonian Seals are infinitely complex. This means you can always add smaller ones depending on what you want. You're limited only by the accuracy of your tools (or, if you're using a computer, the zoom ability of your drawing program). Each circle, no matter how small, should be tangent to the other three. To draw subsequent circles, use the curvatures of the three circles to which they will be tangent in Descartes' Theorem. Then, use the answer (which will be the radius of the new circle) to accurately draw the new circle.
- Note that the Seal we have decided to draw is symmetrical, so the radius of one of the circles is the same as the corresponding circle "through it". However, be aware that not all Apollonian Seals are symmetrical.
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Let's take another example. Let's say that, after drawing the last set of circles, we want to draw circles that are tangent to the third set, to the second and to the outermost large circle. The curvatures of these circles are respectively 3, 2 and -1. We use these numbers in Descartes' Theorem, setting a = -1, b = 2, and c = 3:
- d = a + b + c ± 2 (sqrt (a × b + b × c + c × a))
- d = -1 + 2 + 3 ± 2 (sqrt (-1 × 2 + 2 × 3 + 3 × -1))
- d = -1 + 2 + 3 ± 2 (sqrt (-2 + 6 + -3))
- d = -1 + 2 + 3 ± 2 (sqrt (1))
- d = -1 + 2 + 3 ± 2
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d = 2, 6. We have two answers! However, as we know our new circle will be smaller than any circle it is tangent to, just a curvature
Step 6. (and therefore a radius of 1/6) would make sense.
- The other answer, 2, currently refers to the hypothetical circle on the "other side" of the tangent point of the second and third circles. This "is" tangent to both these circles and the outermost circle, but it should intersect the circles already drawn, so we can ignore it.
Create an Apollonian Gasket Step 10 Step 8. As a challenge, try to make a non-symmetrical Apollonian Seal by changing the size of the second circle
All Apollonian Seals begin the same way - with a large outer circle serving as the edge of the fractal. However, there is no reason why your second circle should have a radius that is half of the first - we did it that way just because it's simple to understand. For fun, start a new Seal with a second circle of a different size. This will lead you to exciting new avenues of exploration.
