How many circles are in a Apollonian gasket?

Published by Charlie Davidson on

How many circles are in a Apollonian gasket?

The table lists the curvatures of the largest circles in the gasket. Only the first three curvatures (of the five displayed in the table) are needed to completely describe each gasket – all other curvatures can be derived from these three….Integral Apollonian gaskets.

Beginning curvatures Symmetry
−15, 32, 32, 33, 65 D1

How to create an Apollonian gasket?

Five Soddy circles. The largest circle and the smallest circle are each other reflections in the yellow circle. Repeating the procedure of making new circles, that are tangent to three other circles, will produce a circle fractal called an Apollonian gasket.

When was the Apollonian gasket discovered?

The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles.

What are Apollonian qualities?

According to Nietzsche, the Apollonian attributes are reason, culture, harmony, and restraint. These are opposed to the Dionysian characteristics of excess, irrationality, lack of discipline, and unbridled passion.

What is a Dionysian experience?

The Dionysian is a force of chaos and destruction, which is the overpowering and alluring chaotic state of wild nature.

Is music Apollonian or Dionysian?

Rational thought is also Apollonian since it is structured and makes distinctions. Drunkenness and madness are Dionysian All forms of enthusiasm and ecstasy are Dionysian. Music is the most Dionysian of the arts, since it appeals directly to man’s instinctive, chaotic emotions and not to his formally reasoning mind.

What are Dionysus virtues?

He had special powers of making wine and causing vines to grow. He could also transform himself into animals such as a bull or a lion. One of his special powers was the ability to drive mortals insane. Dionysus is unique among the Olympic gods in that one of his parents, his mother Semele, was a mortal.

Is music an Apollonian?

What is a Dionysian response?

The Dionysian response indicates “that side of the human personality dominated by feelings, intuition, and freedom from limits: the side that responds emotionally to music as well as to the “force and fury of tragedy.” Apollonian responses seek objective analysis.

Does Dionysus have a wife?

Dionysus takes Ariadne to Olympia and marries her; and he presents her the crown which will later become a star in the sky4.

How are the circles in an Apollonian gasket formed?

An Apollonian Gasket is a type of fractal image that is formed from a collection of ever-shrinking circles contained within a single large circle. Each circle in the Apollonian Gasket is tangent to the adjacent circles – in other words, the circles in the Apollonian Gasket make contact at infinitely small points.

How do you find the radius of a gasket?

Now that we have three circles in our Gasket, we can use Descartes’s Theorem to find the radius of the next circle we’ll draw. Remember that Descartes’s 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 and d is the curvature of the circle tangent to all three.

How did the Apollonian fractal get its name?

Named for the Greek mathematician Apollonius of Perga, this type of fractal can be drawn (by hand or by computer) to reasonable degree of complexity, forming a beautiful, striking image. See Step 1 below to get started.

Categories: Trending