Return to Answer

If from the initial circles $a,b,c,d$ with curvatures $k_a, k_b, k_c, k_d$ and centers $c_a, c_b, c_c, c_d$ you generate four new circles with curvatures:

reflection of $a$: $k_e = 2(k_b+k_c+k_d) - k_a$

reflection of $b$: $k_f = 2(k_c+k_d+k_a) - k_b$

reflection of $c$: $k_g = 2(k_d+k_a+k_b) - k_c$

reflection of $d$: $k_h = 2(k_a+k_b+k_c) - k_d$

Those new circles have centers (complex plane):

$c_e = \frac{2(k_bc_b +k_cc_c +k_dc_d) - k_ac_a}{k_e}$

$c_f = \frac{2(k_cc_c +k_dc_d + k_ac_a) - k_bc_b}{k_f}$

$c_g = \frac{2(k_dc_d +k_ac_a+k_bc_b) - k_cc_c}{k_g}$

$c_h = \frac{2(k_ac_a +k_bc_b +k_cc_c) - k_dc_d}{k_h}$

After the first step, each new circle can generate three more unique circles, e.g., if a circle $e$ is created with $a, b, c, d$ as above then three new circles can be generated with:

$k_i = 2(k_c+k_d+k_e) - k_b$

$k_j = 2(k_b+k_d+k_e) - k_c$

$k_k = 2(k_c+k_b+k_e) - k_d$

$c_i = \frac{2(k_cc_c +k_dc_d +k_ec_e) - k_bc_b}{k_i}$

$c_j = \frac{2(k_bc_b +k_dc_d + k_ec_e) - k_cc_c}{k_j}$

$c_k = \frac{2(k_cc_c +k_bc_b+k_ec_e) - k_dc_d}{k_k}$

...and so on ad infinitum, with the number of circles going up a factor of three for each step.

see: Beyond the Descartes circle theorem

If from the initial circles $a,b,c,d$ with curvatures $k_a, k_b, k_c, k_d$ and centers $c_a, c_b, c_c, c_d$ you generate four new circles with curvatures:

reflection of $a$: $k_e = 2(k_b+k_c+k_d) - k_a$

reflection of $b$: $k_f = 2(k_c+k_d+k_a) - k_b$

reflection of $c$: $k_g = 2(k_d+k_a+k_b) - k_c$

reflection of $d$: $k_h = 2(k_a+k_b+k_c) - k_d$

Those new circles have centers (complex plane):

$c_e = \dfrac{2(k_bc_b +k_cc_c +k_dc_d) - k_ac_a}{k_e}$

$c_f = \dfrac{2(k_cc_c +k_dc_d + k_ac_a) - k_bc_b}{k_f}$

$c_g = \dfrac{2(k_dc_d +k_ac_a+k_bc_b) - k_cc_c}{k_g}$

$c_h = \dfrac{2(k_ac_a +k_bc_b +k_cc_c) - k_dc_d}{k_h}$

After the first step, each new circle can generate three more unique circles, e.g., if a circle $e$ is created with $a, b, c, d$ as above then three new circles can be generated with:

$k_i = 2(k_c+k_d+k_e) - k_b$

$k_j = 2(k_b+k_d+k_e) - k_c$

$k_k = 2(k_c+k_b+k_e) - k_d$

$c_i = \dfrac{2(k_cc_c +k_dc_d +k_ec_e) - k_bc_b}{k_i}$

$c_j = \dfrac{2(k_bc_b +k_dc_d + k_ec_e) - k_cc_c}{k_j}$

$c_k = \dfrac{2(k_cc_c +k_bc_b+k_ec_e) - k_dc_d}{k_k}$

...and so on ad infinitum, with the number of circles going up a factor of three for each step.

see: Beyond the Descartes circle theorem

If from the initial circles $a,b,c,d$ with curvature $k_a, k_b, k_c, k_d$ and centers $c_a, c_b, c_c, c_d$ you generate four new circles with curvatures:

reflection of $a$: $k_e = 2(k_b+k_c+k_d) - k_a$

reflection of $b$: $k_f = 2(k_c+k_d+k_a) - k_b$

reflection of $c$: $k_g = 2(k_d+k_a+k_b) - k_c$

reflection of $d$: $k_h = 2(k_a+k_b+k_c) - k_d$

Those new circles have centers (complex plane):

$c_e = \frac{2(k_bc_b +k_cc_c +k_dc_d) - k_ac_a}{k_e}$

$c_f = \frac{2(k_cc_c +k_dc_d + k_ac_a) - k_bc_b}{k_f}$

$c_g = \frac{2(k_dc_d +k_ac_a+k_bc_b) - k_cc_c}{k_g}$

$c_h = \frac{2(k_ac_a +k_bc_b +k_cc_c) - k_dc_d}{k_h}$

After the first step, each new circle can generate three more unique circles, i.e., if a circle $e$ is created with $a, b, c, d$ as above then three new circles can be generated with:

$k_i = 2(k_c+k_d+k_e) - k_b$

$k_j = 2(k_b+k_d+k_e) - k_c$

$k_k = 2(k_c+k_b+k_e) - k_d$

$c_i = \frac{2(k_cc_c +k_dc_d +k_ec_e) - k_bc_b}{k_i}$

$c_j = \frac{2(k_bc_b +k_dc_d + k_ec_e) - k_cc_c}{k_j}$

$c_k = \frac{2(k_cc_c +k_bc_b+k_ec_e) - k_dc_d}{k_k}$

...and so on ad infinitum, with the number of circles going up a factor of three for each step.

see: Beyond the Descartes circle theorem

If from the initial circles $a,b,c,d$ with curvatures $k_a, k_b, k_c, k_d$ and centers $c_a, c_b, c_c, c_d$ you generate four new circles with curvatures:

reflection of $a$: $k_e = 2(k_b+k_c+k_d) - k_a$

reflection of $b$: $k_f = 2(k_c+k_d+k_a) - k_b$

reflection of $c$: $k_g = 2(k_d+k_a+k_b) - k_c$

reflection of $d$: $k_h = 2(k_a+k_b+k_c) - k_d$

Those new circles have centers (complex plane):

$c_e = \frac{2(k_bc_b +k_cc_c +k_dc_d) - k_ac_a}{k_e}$

$c_f = \frac{2(k_cc_c +k_dc_d + k_ac_a) - k_bc_b}{k_f}$

$c_g = \frac{2(k_dc_d +k_ac_a+k_bc_b) - k_cc_c}{k_g}$

$c_h = \frac{2(k_ac_a +k_bc_b +k_cc_c) - k_dc_d}{k_h}$

After the first step, each new circle can generate three more unique circles, e.g., if a circle $e$ is created with $a, b, c, d$ as above then three new circles can be generated with:

$k_i = 2(k_c+k_d+k_e) - k_b$

$k_j = 2(k_b+k_d+k_e) - k_c$

$k_k = 2(k_c+k_b+k_e) - k_d$

$c_i = \frac{2(k_cc_c +k_dc_d +k_ec_e) - k_bc_b}{k_i}$

$c_j = \frac{2(k_bc_b +k_dc_d + k_ec_e) - k_cc_c}{k_j}$

$c_k = \frac{2(k_cc_c +k_bc_b+k_ec_e) - k_dc_d}{k_k}$

...and so on ad infinitum, with the number of circles going up a factor of three for each step.

see: Beyond the Descartes circle theorem

If from the initial circles with curvature $k_a, k_b, k_c, k_d$ and centers $c_a, c_b, c_c, c_d$ you generate four new circles with curvatures:

reflection of a: $k_e = 2(k_b+k_c+k_d) - k_a$

reflection of b: $k_f = 2(k_c+k_d+k_a) - k_b$

reflection of c: $k_g = 2(k_d+k_a+k_b) - k_c$

reflection of d: $k_h = 2(k_a+k_b+k_c) - k_d$

Those circles have centers (complex plane):

$c_e = \frac{2(k_bc_b +k_cc_c +k_dc_d) - k_ac_a}{k_e}$

$c_f = \frac{2(k_cc_c +k_dc_d + k_ac_a) - k_bc_b}{k_f}$

$c_g = \frac{2(k_dc_d +k_ac_a+k_bc_b) - k_cc_c}{k_g}$

$c_h = \frac{2(k_ac_a +k_bc_b +k_cc_c) - k_dc_d}{k_h}$

After the first step, each new circle can generate three more unique circles, i.e., if a circle $e$ is created with $a, b, c, d$ as above then three new circles can be generated with:

$k_i = 2(k_c+k_d+k_e) - k_b$

$k_j = 2(k_b+k_d+k_e) - k_c$

$k_k = 2(k_c+k_b+k_e) - k_d$

$c_i = \frac{2(k_cc_c +k_dc_d +k_ec_e) - k_bc_b}{k_i}$

$c_j = \frac{2(k_bc_b +k_dc_d + k_ec_e) - k_cc_c}{k_j}$

$c_k = \frac{2(k_cc_c +k_bc_b+k_ec_e) - k_dc_d}{k_k}$

...and so on ad infinitum, with the number of circles going up a factor of three for each step.

see: Beyond the Descartes circle theorem

If from the initial circles $a,b,c,d$ with curvature $k_a, k_b, k_c, k_d$ and centers $c_a, c_b, c_c, c_d$ you generate four new circles with curvatures:

reflection of $a$: $k_e = 2(k_b+k_c+k_d) - k_a$

reflection of $b$: $k_f = 2(k_c+k_d+k_a) - k_b$

reflection of $c$: $k_g = 2(k_d+k_a+k_b) - k_c$

reflection of $d$: $k_h = 2(k_a+k_b+k_c) - k_d$

Those new circles have centers (complex plane):

$c_e = \frac{2(k_bc_b +k_cc_c +k_dc_d) - k_ac_a}{k_e}$

$c_f = \frac{2(k_cc_c +k_dc_d + k_ac_a) - k_bc_b}{k_f}$

$c_g = \frac{2(k_dc_d +k_ac_a+k_bc_b) - k_cc_c}{k_g}$

$c_h = \frac{2(k_ac_a +k_bc_b +k_cc_c) - k_dc_d}{k_h}$

After the first step, each new circle can generate three more unique circles, i.e., if a circle $e$ is created with $a, b, c, d$ as above then three new circles can be generated with:

$k_i = 2(k_c+k_d+k_e) - k_b$

$k_j = 2(k_b+k_d+k_e) - k_c$

$k_k = 2(k_c+k_b+k_e) - k_d$

$c_i = \frac{2(k_cc_c +k_dc_d +k_ec_e) - k_bc_b}{k_i}$

$c_j = \frac{2(k_bc_b +k_dc_d + k_ec_e) - k_cc_c}{k_j}$

$c_k = \frac{2(k_cc_c +k_bc_b+k_ec_e) - k_dc_d}{k_k}$

...and so on ad infinitum, with the number of circles going up a factor of three for each step.

see: Beyond the Descartes circle theorem