apollonian circles: why are radius and center dual?
This figure suggests the radii and centers (regarded as complex numbers) of the Soddy circles satisfy the same equation:
$$ a^2 + b^2 + c^2 + d^2 = frac{1}{2} (a + b + c + d)^2$$
How can the circle and radius be dual in this particular sangaku problem?
http://dl.dropbox.com/u/17949100/soddy.png
geometry euclidean-geometry sangaku
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63464
asked Apr 10 '13 at 18:51
cactus314
15.3k42169
add a comment |
This figure suggests the radii and centers (regarded as complex numbers) of the Soddy circles satisfy the same equation:
$$ a^2 + b^2 + c^2 + d^2 = frac{1}{2} (a + b + c + d)^2$$
How can the circle and radius be dual in this particular sangaku problem?
http://dl.dropbox.com/u/17949100/soddy.png
geometry euclidean-geometry sangaku
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63464
asked Apr 10 '13 at 18:51
cactus314
15.3k42169
I don't understand what the question is. Are you asking for a proof of that equation (for both centers and radii)? Have you looked at "Wilks et al."?
– Hew Wolff
Apr 10 '13 at 19:01
@HewWolff, two articles on Apollonian circles in the present Bulletin, see link in my "answer." Hmmm, one P, two L's
– Will Jagy
Apr 10 '13 at 20:58
The link is dead, could you insert the image in the question?
– Jean-Claude Arbaut
Nov 23 at 6:41
add a comment |
This figure suggests the radii and centers (regarded as complex numbers) of the Soddy circles satisfy the same equation:
$$ a^2 + b^2 + c^2 + d^2 = frac{1}{2} (a + b + c + d)^2$$
How can the circle and radius be dual in this particular sangaku problem?
http://dl.dropbox.com/u/17949100/soddy.png
geometry euclidean-geometry sangaku
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63464
asked Apr 10 '13 at 18:51
cactus314
15.3k42169
This figure suggests the radii and centers (regarded as complex numbers) of the Soddy circles satisfy the same equation:
$$ a^2 + b^2 + c^2 + d^2 = frac{1}{2} (a + b + c + d)^2$$
How can the circle and radius be dual in this particular sangaku problem?
http://dl.dropbox.com/u/17949100/soddy.png
geometry euclidean-geometry sangaku
geometry euclidean-geometry sangaku
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63464
asked Apr 10 '13 at 18:51
cactus314
15.3k42169
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63464
asked Apr 10 '13 at 18:51
cactus314
15.3k42169
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63464
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63464
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63464
14.7k63464
asked Apr 10 '13 at 18:51
cactus314
15.3k42169
asked Apr 10 '13 at 18:51
cactus314
15.3k42169
asked Apr 10 '13 at 18:51
cactus314
15.3k42169
15.3k42169
I don't understand what the question is. Are you asking for a proof of that equation (for both centers and radii)? Have you looked at "Wilks et al."?
– Hew Wolff
Apr 10 '13 at 19:01
@HewWolff, two articles on Apollonian circles in the present Bulletin, see link in my "answer." Hmmm, one P, two L's
– Will Jagy
Apr 10 '13 at 20:58
The link is dead, could you insert the image in the question?
– Jean-Claude Arbaut
Nov 23 at 6:41
add a comment |
I don't understand what the question is. Are you asking for a proof of that equation (for both centers and radii)? Have you looked at "Wilks et al."?
– Hew Wolff
Apr 10 '13 at 19:01
@HewWolff, two articles on Apollonian circles in the present Bulletin, see link in my "answer." Hmmm, one P, two L's
– Will Jagy
Apr 10 '13 at 20:58
The link is dead, could you insert the image in the question?
– Jean-Claude Arbaut
Nov 23 at 6:41
I don't understand what the question is. Are you asking for a proof of that equation (for both centers and radii)? Have you looked at "Wilks et al."?
– Hew Wolff
Apr 10 '13 at 19:01
I don't understand what the question is. Are you asking for a proof of that equation (for both centers and radii)? Have you looked at "Wilks et al."?
– Hew Wolff
Apr 10 '13 at 19:01
@HewWolff, two articles on Apollonian circles in the present Bulletin, see link in my "answer." Hmmm, one P, two L's
– Will Jagy
Apr 10 '13 at 20:58
@HewWolff, two articles on Apollonian circles in the present Bulletin, see link in my "answer." Hmmm, one P, two L's
– Will Jagy
Apr 10 '13 at 20:58
The link is dead, could you insert the image in the question?
– Jean-Claude Arbaut
Nov 23 at 6:41
The link is dead, could you insert the image in the question?
– Jean-Claude Arbaut
Nov 23 at 6:41
add a comment |
1 Answer
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Note that there are two new articles on Apollonian packings, downloadable for free, at BULLETIN
Both bibliographies list four articles, from 2003 to 2006, by Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan. All four titles begin Apollonian circle packings: as mentioned by Hew Wolff
answered Apr 10 '13 at 19:42
Will Jagy
101k599199
add a comment |
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Note that there are two new articles on Apollonian packings, downloadable for free, at BULLETIN
Both bibliographies list four articles, from 2003 to 2006, by Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan. All four titles begin Apollonian circle packings: as mentioned by Hew Wolff
answered Apr 10 '13 at 19:42
Will Jagy
101k599199
add a comment |
Note that there are two new articles on Apollonian packings, downloadable for free, at BULLETIN
Both bibliographies list four articles, from 2003 to 2006, by Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan. All four titles begin Apollonian circle packings: as mentioned by Hew Wolff
answered Apr 10 '13 at 19:42
Will Jagy
101k599199
add a comment |
Note that there are two new articles on Apollonian packings, downloadable for free, at BULLETIN
Both bibliographies list four articles, from 2003 to 2006, by Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan. All four titles begin Apollonian circle packings: as mentioned by Hew Wolff
answered Apr 10 '13 at 19:42
Will Jagy
101k599199
Note that there are two new articles on Apollonian packings, downloadable for free, at BULLETIN
Both bibliographies list four articles, from 2003 to 2006, by Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan. All four titles begin Apollonian circle packings: as mentioned by Hew Wolff
answered Apr 10 '13 at 19:42
Will Jagy
101k599199
edited Apr 10 '13 at 20:59
answered Apr 10 '13 at 19:42
Will Jagy
101k599199
answered Apr 10 '13 at 19:42
Will Jagy
101k599199
answered Apr 10 '13 at 19:42
Will Jagy
101k599199
101k599199
add a comment |
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I don't understand what the question is. Are you asking for a proof of that equation (for both centers and radii)? Have you looked at "Wilks et al."?
– Hew Wolff
Apr 10 '13 at 19:01
@HewWolff, two articles on Apollonian circles in the present Bulletin, see link in my "answer." Hmmm, one P, two L's
– Will Jagy
Apr 10 '13 at 20:58
The link is dead, could you insert the image in the question?
– Jean-Claude Arbaut
Nov 23 at 6:41