Difference between revisions of "Problem of Apollonius"
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<h1> A More Mathematical Explanation</h1> | <h1> A More Mathematical Explanation</h1> | ||
− | There are many different ways of solving the problem of Apollonius. The few that are easiest to understand include using an algebraic method or an [[Inversion|inverse geometry]] method. | + | {{hide|1=There are many different ways of solving the problem of Apollonius. The few that are easiest to understand include using an algebraic method or an [[Inversion|inverse geometry]] method. |
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=== Algebraic Method === | === Algebraic Method === | ||
− | + | This method only uses math up to the level of understanding quadratic equations. We will proceed by setting up a system of quadratic equations and solving for the radius, r, of the unknown circle. | |
We start by labeling the center of each of the given circles <math>(x</math><sub>1</sub><math>,y</math><sub>1</sub><math>)</math>, <math>(x</math><sub>2</sub><math>,y</math><sub>2</sub><math>)</math>, and <math>(x</math><sub>3</sub><math>,y</math><sub>3</sub><math>)</math>. We will call the center of the unknown circle <math>(x,y)</math>. <math>r</math><sub>1</sub>,<math>r</math><sub>2</sub>, and <math>r</math><sub>3</sub> are the different radii of each of the given circles. | We start by labeling the center of each of the given circles <math>(x</math><sub>1</sub><math>,y</math><sub>1</sub><math>)</math>, <math>(x</math><sub>2</sub><math>,y</math><sub>2</sub><math>)</math>, and <math>(x</math><sub>3</sub><math>,y</math><sub>3</sub><math>)</math>. We will call the center of the unknown circle <math>(x,y)</math>. <math>r</math><sub>1</sub>,<math>r</math><sub>2</sub>, and <math>r</math><sub>3</sub> are the different radii of each of the given circles. | ||
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Repeating this process over and over again with each set of three mutually tangent circles will create the Apollonian gasket.}} | Repeating this process over and over again with each set of three mutually tangent circles will create the Apollonian gasket.}} | ||
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Revision as of 14:43, 25 June 2012
Apollonian Gasket |
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Apollonian Gasket
- This an example of a fractal that can be created by repeatedly solving the Problem of Apollonius.
Contents
Basic Description
Apollonius of Perga posed and solved this problem in his work called Tangencies. Sadly, Tangencies has been lost, and only a report of his work by Pappus of Alexandria is left. Since then, other mathematicians, such as Isaac Newton and Descartes, have been able to recreate his results and discover new ways of solving this interesting problem.
The problem usually has eight different solution circles that exist that are tangent to the given three circles in a plane. The given circles must not be tangent to each other, overlapping, or contained within one another for all eight solutions to exist.
Given three points, the problem only has one solution. In the cases of one line and two points; two lines and one point; and one circle and two points, the problem has two solutions. Four solutions exist for the cases of three lines; one circle, one line, and one point; and two circles and one point. There are eight solutions for the cases of two circles and one line; and one circle and two lines, in addition to the three circle problem.
A More Mathematical Explanation
References
Math Pages, Apollonius' Tangency Problem
MathWorld, Apollonius' Problem
Wikibooks , Apollonian fractals
Mathematica Programs
Anna created several programs that solve the problem of Apollonius in the case of three non-tangent, non-intersecting circles that the user inputs. The programs were written in Mathematica 7, but are likely compatible with Mathematica 5 and/or 6.
The first program automatically plots all eight solutions. Click here to download this program
The second program allows the user to choose which solutions to plot in groups of two. Click here to download this program
Ideas for the future
AnnaP has a mathematica program that solves the problem for three non intersecting circles. Turning this code into an applet where people could input three circles would be really cool! Email Anna to get the mathematica script if you want to work on this.
Teaching Materials
- There are currently no teaching materials for this page. Add teaching materials.
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.