Pappus nested circles

NakulG · 2014-09-02 05:40:27

Hi,
I am trying to teach my son to create pappus nested circles as part of a class project.
I need help in terms of formulas that can enable me to construct these on a chart using scale, compass, etc.
For example, how do I find the centre and radius of a circle that is touching 3 or more other circles, I need to know the formulas and steps.

This is urgent.

Thanks in advance.

Nakul

bobbym · 2014-09-02 05:46:20

Hi;

Soddy circles?

NakulG · 2014-09-02 05:52:51

Please see this link.
http://mathworld.wolfram.com/PappusChain.html
How do I explain to my son how these formulas are derived.
Thanks

bobbym · 2014-09-02 05:54:08

How old is he?

NakulG · 2014-09-02 05:55:07

13+

bobbym · 2014-09-02 05:56:55

That is quite difficult stuff. Mathworld is typically hard. What does he not understand or rather what do you not understand?

NakulG · 2014-09-02 05:59:10

Starting from the first circle, how do I get the centre and radius of circle2 and then from that how do I get the centre and radius of 3rd and so on and so forth.

bobbym · 2014-09-02 06:03:12

Are you looking for the red circles or the grey?

I will be back in about a half hour.

NakulG · 2014-09-02 06:06:43

Both red and grey, but they come in order, first the greys, and then the nested red and then the second grey and then the nested red and so on soforth.

bobbym · 2014-09-02 06:33:33

Do you use some type of software like geogebra?

This page does not work on my browser but it might on yours:

http://www.geogebra.org/en/examples/fri … chain.html

did that help you?

NakulG · 2014-09-02 08:10:29

Yes I tried geogebra, but it is again a hit or trial method. It is not precise. For it to be precise, (exactly tangential / kissing circles), it needs to be created by deriving the centre and radius.
Thanks again.

Last edited by NakulG (2014-09-02 08:11:24)

bobbym · 2014-09-02 08:33:38

You can not draw them exact with a compass either.

The formulas for the circles and arbelos are on that page. You will have to run the recurrences until all the circles are drawn. That means computing them using those formulas. This will be tedious and error prone.

NakulG · 2014-09-02 12:32:13

Hard to believe, but I will continue to explore.
Thanks

Last edited by NakulG (2014-09-02 12:37:03)

NakulG · 2014-09-13 04:33:15

I tried converting the formulas from Wolfram Alpha into a spreadsheet. It does not seem to work, I am not sure why? Can you help me here?
I can send you the spreadsheet, how do i upload the spreadsheet here ...?

bobbym · 2014-09-13 04:37:49

What formula? Please post it or screenshot it.

NakulG · 2014-09-13 04:53:40

same formulas for x, y & r for P and T, that are in wolfram alpha page on pappus chain...

http://mathworld.wolfram.com/PappusChain.html

I can send the whole spreadsheet with the chart etc.

NakulG · 2014-09-13 04:59:14

i cant figure out how to attach the file / upload image ....

bobbym · 2014-09-13 05:01:17

Do you have the link? Post that and I will post the picture for you.

NakulG · 2014-09-13 05:03:31

Hope this helps

bobbym · 2014-09-13 05:08:45

Okay, the first one:

how are you trying to enter it?

NakulG · 2014-09-13 05:12:00

AB r Ratio
AC AB 1 1 1 1 1 1 1 1 1
AC 2 3 4 5 6 7 8 9 10
Pn Xn Ratio
n 0.50 0.33 0.25 0.20 0.17 0.14 0.13 0.11 0.10
0 0.75000 0.66667 0.62500 0.60000 0.58333 0.57143 0.56250 0.55556 0.55000
1 0.50000 0.28571 0.19231 0.14286 0.11290 0.09302 0.07895 0.06849 0.06044
2 0.25000 0.10526 0.06250 0.04348 0.03302 0.02649 0.02206 0.01887 0.01647
3 0.13636 0.05128 0.02941 0.02013 0.01515 0.01208 0.01002 0.00855 0.00744
4 0.08333 0.02985 0.01689 0.01149 0.00862 0.00686 0.00568 0.00484 0.00421
5 0.05556 0.01942 0.01092 0.00741 0.00555 0.00441 0.00365 0.00311 0.00270
6 0.03947 0.01361 0.00762 0.00516 0.00386 0.00307 0.00254 0.00216 0.00188
7 0.02941 0.01005 0.00562 0.00380 0.00284 0.00226 0.00187 0.00159 0.00138
8 0.02273 0.00772 0.00431 0.00292 0.00218 0.00173 0.00143 0.00122 0.00106
9 0.01807 0.00612 0.00341 0.00231 0.00172 0.00137 0.00113 0.00096 0.00084
10 0.01471 0.00496 0.00277 0.00187 0.00140 0.00111 0.00092 0.00078 0.00068

NakulG · 2014-09-13 05:14:03

Yn Ratio
n 0.50 0.33 0.25 0.20 0.17 0.14 0.13 0.11 0.10
0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
1 0.33333 0.28571 0.23077 0.19048 0.16129 0.13953 0.12281 0.10959 0.09890
2 0.33333 0.21053 0.15000 0.11594 0.09434 0.07947 0.06863 0.06038 0.05389
3 0.27273 0.15385 0.10588 0.08054 0.06494 0.05438 0.04677 0.04103 0.03654
4 0.22222 0.11940 0.08108 0.06130 0.04926 0.04117 0.03535 0.03098 0.02757
5 0.18519 0.09709 0.06550 0.04938 0.03962 0.03308 0.02839 0.02486 0.02211
6 0.15789 0.08163 0.05488 0.04131 0.03311 0.02763 0.02370 0.02075 0.01846
7 0.13725 0.07035 0.04719 0.03549 0.02843 0.02372 0.02034 0.01781 0.01583
8 0.12121 0.06178 0.04138 0.03110 0.02491 0.02077 0.01781 0.01559 0.01386
9 0.10843 0.05505 0.03683 0.02767 0.02216 0.01847 0.01584 0.01386 0.01233
10 0.09804 0.04963 0.03319 0.02492 0.01995 0.01663 0.01426 0.01248 0.01110

Rn Ratio
n 0.50 0.33 0.25 0.20 0.17 0.14 0.13 0.11 0.10
0 0.25000 0.33333 0.37500 0.40000 0.41667 0.42857 0.43750 0.44444 0.45000
1 0.16667 0.14286 0.11538 0.09524 0.08065 0.06977 0.06140 0.05479 0.04945
2 0.08333 0.05263 0.03750 0.02899 0.02358 0.01987 0.01716 0.01509 0.01347
3 0.04545 0.02564 0.01765 0.01342 0.01082 0.00906 0.00780 0.00684 0.00609
4 0.02778 0.01493 0.01014 0.00766 0.00616 0.00515 0.00442 0.00387 0.00345
5 0.01852 0.00971 0.00655 0.00494 0.00396 0.00331 0.00284 0.00249 0.00221
6 0.01316 0.00680 0.00457 0.00344 0.00276 0.00230 0.00198 0.00173 0.00154
7 0.00980 0.00503 0.00337 0.00253 0.00203 0.00169 0.00145 0.00127 0.00113
8 0.00758 0.00386 0.00259 0.00194 0.00156 0.00130 0.00111 0.00097 0.00087
9 0.00602 0.00306 0.00205 0.00154 0.00123 0.00103 0.00088 0.00077 0.00068
10 0.00490 0.00248 0.00166 0.00125 0.00100 0.00083 0.00071 0.00062 0.00055

Tn Xn Ratio
n 0.50 0.33 0.25 0.20 0.17 0.14 0.13 0.11 0.10
0 0.50000 0.32353 0.23770 0.18750 0.15468 0.13158 0.11446 0.10127 0.09079
1 0.50000 0.32353 0.23770 0.18750 0.15468 0.13158 0.11446 0.10127 0.09079
2 0.32609 0.16667 0.10902 0.08036 0.06342 0.05230 0.04446 0.03865 0.03417
3 0.19231 0.08462 0.05235 0.03750 0.02909 0.02372 0.02000 0.01728 0.01520
4 0.11905 0.04867 0.02941 0.02083 0.01606 0.01303 0.01096 0.00945 0.00830
5 0.07895 0.03107 0.01857 0.01308 0.01005 0.00814 0.00684 0.00589 0.00517
6 0.05556 0.02140 0.01271 0.00893 0.00685 0.00554 0.00465 0.00400 0.00351
7 0.04098 0.01558 0.00922 0.00647 0.00496 0.00401 0.00336 0.00289 0.00254
8 0.03138 0.01183 0.00698 0.00489 0.00375 0.00303 0.00254 0.00218 0.00192
9 0.02475 0.00927 0.00547 0.00383 0.00293 0.00237 0.00198 0.00171 0.00150
10 0.02000 0.00746 0.00439 0.00307 0.00235 0.00190 0.00159 0.00137 0.00120

Yn Ratio
n 0.50 0.33 0.25 0.20 0.17 0.14 0.13 0.11 0.10
0 -0.13333 -0.11765 -0.09836 -0.08333 -0.07194 -0.06316 -0.05622 -0.05063 -0.04604
1 0.13333 0.11765 0.09836 0.08333 0.07194 0.06316 0.05622 0.05063 0.04604
2 0.26087 0.18182 0.13534 0.10714 0.08850 0.07531 0.06552 0.05797 0.05197
3 0.25641 0.15385 0.10830 0.08333 0.06766 0.05693 0.04912 0.04320 0.03854
4 0.22222 0.12389 0.08519 0.06481 0.05228 0.04380 0.03768 0.03306 0.02945
5 0.18947 0.10169 0.06914 0.05233 0.04208 0.03518 0.03022 0.02649 0.02358
6 0.16296 0.08560 0.05784 0.04365 0.03504 0.02927 0.02513 0.02201 0.01958
7 0.14208 0.07365 0.04959 0.03736 0.02996 0.02501 0.02146 0.01879 0.01672
8 0.12552 0.06452 0.04333 0.03261 0.02614 0.02181 0.01871 0.01638 0.01457
9 0.11221 0.05734 0.03845 0.02891 0.02316 0.01932 0.01657 0.01451 0.01290
10 0.10133 0.05156 0.03453 0.02596 0.02079 0.01734 0.01487 0.01302 0.01157

Rn Ratio
n 0.50 0.33 0.25 0.20 0.17 0.14 0.13 0.11 0.10
0 0.03333 0.02941 0.02459 0.02083 0.01799 0.01579 0.01406 0.01266 0.01151
1 0.03333 0.02941 0.02459 0.02083 0.01799 0.01579 0.01406 0.01266 0.01151
2 0.02174 0.01515 0.01128 0.00893 0.00737 0.00628 0.00546 0.00483 0.00433
3 0.01282 0.00769 0.00542 0.00417 0.00338 0.00285 0.00246 0.00216 0.00193
4 0.00794 0.00442 0.00304 0.00231 0.00187 0.00156 0.00135 0.00118 0.00105
5 0.00526 0.00282 0.00192 0.00145 0.00117 0.00098 0.00084 0.00074 0.00065
6 0.00370 0.00195 0.00131 0.00099 0.00080 0.00067 0.00057 0.00050 0.00045
7 0.00273 0.00142 0.00095 0.00072 0.00058 0.00048 0.00041 0.00036 0.00032
8 0.00209 0.00108 0.00072 0.00054 0.00044 0.00036 0.00031 0.00027 0.00024
9 0.00165 0.00084 0.00057 0.00043 0.00034 0.00028 0.00024 0.00021 0.00019
10 0.00133 0.00068 0.00045 0.00034 0.00027 0.00023 0.00020 0.00017 0.00015

Ratio 0.5
Pn Xn Yn Area n Rn
n 0 0.75000 0.00000 1.76715 0.25000 0.56250 0.00000 0.56250 0.75000
1 0.50000 0.33333 1.13446 0.16667 0.25000 0.11111 0.36111 0.60093
2 0.25000 0.33333 0.54542 0.08333 0.06250 0.11111 0.17361 0.41667
3 0.13636 0.27273 0.29209 0.04545 0.01860 0.07438 0.09298 0.30492
4 0.08333 0.22222 0.17696 0.02778 0.00694 0.04938 0.05633 0.23733
5 0.05556 0.18519 0.11743 0.01852 0.00309 0.03429 0.03738 0.19334
6 0.03947 0.15789 0.08322 0.01316 0.00156 0.02493 0.02649 0.16275
7 0.02941 0.13725 0.06190 0.00980 0.00087 0.01884 0.01970 0.14037
8 0.02273 0.12121 0.04778 0.00758 0.00052 0.01469 0.01521 0.12332
9 0.01807 0.10843 0.03796 0.00602 0.00033 0.01176 0.01208 0.10993
10 0.01471 0.09804 0.03088 0.00490 0.00022 0.00961 0.00983 0.09914
Tn 0 0.50000 -0.13333 0.84125 0.03333 0.25000 0.01778 0.26778 0.51747
1 0.50000 0.13333 0.84125 0.03333 0.25000 0.01778 0.26778 0.51747
2 0.32609 0.26087 0.54785 0.02174 0.10633 0.06805 0.17439 0.41760
3 0.19231 0.25641 0.32273 0.01282 0.03698 0.06575 0.10273 0.32051
4 0.11905 0.22222 0.19966 0.00794 0.01417 0.04938 0.06356 0.25210
5 0.07895 0.18947 0.13236 0.00526 0.00623 0.03590 0.04213 0.20526
6 0.05556 0.16296 0.09313 0.00370 0.00309 0.02656 0.02964 0.17217
7 0.04098 0.14208 0.06869 0.00273 0.00168 0.02019 0.02187 0.14787
8 0.03138 0.12552 0.05259 0.00209 0.00098 0.01576 0.01674 0.12939
9 0.02475 0.11221 0.04148 0.00165 0.00061 0.01259 0.01320 0.11491
10 0.02000 0.10133 0.03352 0.00133 0.00040 0.01027 0.01067 0.10329

NakulG · 2014-09-13 05:17:17

I am taking r 0.5 for creating the chart and for n varying from 1 to 10.

bobbym · 2014-09-13 05:19:36

All that is for the first equation?

NakulG · 2014-09-13 05:22:35

Apologies for pushing so much data thru.

No.

for the first equation the values are as follows (r varies from 0.5 to 0.1) and n varies for 0 to 10
Pn Xn Ratio (r)
n 0.50 0.33 0.25 0.20 0.17 0.14 0.13 0.11 0.10
0 0.75000 0.66667 0.62500 0.60000 0.58333 0.57143 0.56250 0.55556 0.55000
1 0.50000 0.28571 0.19231 0.14286 0.11290 0.09302 0.07895 0.06849 0.06044
2 0.25000 0.10526 0.06250 0.04348 0.03302 0.02649 0.02206 0.01887 0.01647
3 0.13636 0.05128 0.02941 0.02013 0.01515 0.01208 0.01002 0.00855 0.00744
4 0.08333 0.02985 0.01689 0.01149 0.00862 0.00686 0.00568 0.00484 0.00421
5 0.05556 0.01942 0.01092 0.00741 0.00555 0.00441 0.00365 0.00311 0.00270
6 0.03947 0.01361 0.00762 0.00516 0.00386 0.00307 0.00254 0.00216 0.00188
7 0.02941 0.01005 0.00562 0.00380 0.00284 0.00226 0.00187 0.00159 0.00138
8 0.02273 0.00772 0.00431 0.00292 0.00218 0.00173 0.00143 0.00122 0.00106
9 0.01807 0.00612 0.00341 0.00231 0.00172 0.00137 0.00113 0.00096 0.00084
10 0.01471 0.00496 0.00277 0.00187 0.00140 0.00111 0.00092 0.00078 0.00068

Math Is Fun Forum

#1 2014-09-02 05:40:27

Pappus nested circles

#2 2014-09-02 05:46:20

Re: Pappus nested circles

#3 2014-09-02 05:52:51

Re: Pappus nested circles

#4 2014-09-02 05:54:08

Re: Pappus nested circles

#5 2014-09-02 05:55:07

Re: Pappus nested circles

#6 2014-09-02 05:56:55

Re: Pappus nested circles

#7 2014-09-02 05:59:10

Re: Pappus nested circles

#8 2014-09-02 06:03:12

Re: Pappus nested circles

#9 2014-09-02 06:06:43

Re: Pappus nested circles

#10 2014-09-02 06:33:33

Re: Pappus nested circles

#11 2014-09-02 08:10:29

Re: Pappus nested circles

#12 2014-09-02 08:33:38

Re: Pappus nested circles

#13 2014-09-02 12:32:13

Re: Pappus nested circles

#14 2014-09-13 04:33:15

Re: Pappus nested circles

#15 2014-09-13 04:37:49

Re: Pappus nested circles

#16 2014-09-13 04:53:40

Re: Pappus nested circles

#17 2014-09-13 04:59:14

Re: Pappus nested circles

#18 2014-09-13 05:01:17

Re: Pappus nested circles

#19 2014-09-13 05:03:31

Re: Pappus nested circles

#20 2014-09-13 05:08:45

Re: Pappus nested circles

#21 2014-09-13 05:12:00

Re: Pappus nested circles

#22 2014-09-13 05:14:03

Re: Pappus nested circles

#23 2014-09-13 05:17:17

Re: Pappus nested circles

#24 2014-09-13 05:19:36

Re: Pappus nested circles

#25 2014-09-13 05:22:35

Re: Pappus nested circles

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