Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Pappus nested circles

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

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

Hi;

Soddy circles?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

How old is he?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

13+

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

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


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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.

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

Are you looking for the red circles or the grey?

PappusTangentChain_800.gif

I will be back in about a half hour.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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.

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

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?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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)

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

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.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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

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

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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 ...?

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

What formula? Please post it or screenshot it.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

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


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

Hope this helps

Inline14.gif

Inline17.gif

Inline20.gif

Inline60.gif

Inline63.gif

Inline66.gif

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

Okay, the first one:

Inline14.gif

how are you trying to enter it?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Pappus nested circles

All that is for the first equation?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

NakulG
Member
Registered: 2014-09-02
Posts: 186

Re: Pappus nested circles

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

Offline

Board footer

Powered by FluxBB