apollonian circles: why are radius and center dual?
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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
14.7k63363
asked Apr 10 '13 at 18:51
cactus314
15.2k42068
add a comment |
up vote
2
down vote
favorite
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
14.7k63363
asked Apr 10 '13 at 18:51
cactus314
15.2k42068
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 |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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
14.7k63363
asked Apr 10 '13 at 18:51
cactus314
15.2k42068
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
14.7k63363
asked Apr 10 '13 at 18:51
cactus314
15.2k42068
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63363
asked Apr 10 '13 at 18:51
cactus314
15.2k42068
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63363
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63363
edited Nov 23 at 6:41
Jean-Claude Arbaut
14.7k63363
14.7k63363
asked Apr 10 '13 at 18:51
cactus314
15.2k42068
asked Apr 10 '13 at 18:51
cactus314
15.2k42068
asked Apr 10 '13 at 18:51
cactus314
15.2k42068
15.2k42068
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
101k598198
add a comment |
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1 Answer
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oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
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
101k598198
add a comment |
up vote
1
down vote
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
101k598198
add a comment |
up vote
1
down vote
up vote
1
down vote
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
101k598198
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
101k598198
edited Apr 10 '13 at 20:59
answered Apr 10 '13 at 19:42
Will Jagy
101k598198
answered Apr 10 '13 at 19:42
Will Jagy
101k598198
answered Apr 10 '13 at 19:42
Will Jagy
101k598198
101k598198
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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