Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2004
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2004

[Date Index] [Thread Index] [Author Index]

Search the Archive

Use of a rational tiling group in sl(2,R) to get a 3d surface

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49296] Use of a rational tiling group in sl(2,R) to get a 3d surface
  • From: Roger Bagula <tftn at earthlink.net>
  • Date: Mon, 12 Jul 2004 02:11:37 -0400 (EDT)
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com

This method was suggested by the Bryant cousin surface.

It gives an hyperboloid of one sheet that is very like a catenoid in shape.
A determinant one group is assumed through out.
This result is very different that the intent of Lagarias
in terms of a upper half plane rational tiling.
Respectfully, Roger L. Bagula

tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/

(***********************************************************************

                    Mathematica-Compatible Notebook

This notebook can be used on any computer system with Mathematica 3.0,
MathReader 3.0, or any compatible application. The data for the notebook 
starts with the line of stars above.

To get the notebook into a Mathematica-compatible application, do one of 
the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing the 
word CacheID, otherwise Mathematica-compatible applications may try to 
use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info at wolfram.com
  phone: +1-217-398-0700 (U.S.)

Notebook reader applications are available free of charge from 
Wolfram Research.
***********************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[      8382,        263]*)
(*NotebookOutlinePosition[      9317,        293]*)
(*  CellTagsIndexPosition[      9273,        289]*)
(*WindowFrame->Normal*)



Notebook[{
Cell[BoxData[
    \( (*\ 
      marked\ tiles\ sl \((2, r)\)\ rational\ 
        \(group : \ 
          \(page291 : \ 
            A\ walk\ along\ the\ branches\ of\ the\ extended\ Farey\ Tree\ by
              \ Lagarias\ and\ Tresser\)\)*) \)], "Input"],

Cell[BoxData[
    \( (*\ By\ Roger\ L . \ Bagula\ 11\ July\ 2004  \[Copyright]*) \)], 
  "Input"],

Cell[BoxData[
    \( (*\ defined\ as\ Det[a] = \(Det[b] = \(Det[c] = 1\)\)*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(a = {{p1, p0}, {q1, q0}}\)], "Input"],

Cell[BoxData[
    \({{p1, p0}, {q1, q0}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(b = {{\(-p0\), p0 + p1}, {\(-q0\), q0 + q1}}\)], "Input"],

Cell[BoxData[
    \({{\(-p0\), p0 + p1}, {\(-q0\), q0 + q1}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(c = {{p0 + p1, p1}, {q0 + q1, \(-q1\)}}\)], "Input"],

Cell[BoxData[
    \({{p0 + p1, p1}, {q0 + q1, \(-q1\)}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(a1 = MatrixPower[a, \(-1\)]\)], "Input"],

Cell[BoxData[
    \({{q0\/\(p1\ q0 - p0\ q1\), \(-\(p0\/\(p1\ q0 - p0\ q1\)\)\)}, {
        \(-\(q1\/\(p1\ q0 - p0\ q1\)\)\), p1\/\(p1\ q0 - p0\ q1\)}}\)], 
  "Output"]
}, Open  ]],

Cell[BoxData[
    \( (*\ group\ definition\ in\ the\ variables {x, y, z}*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(g = x*a + y*b + z*c\)], "Input"],

Cell[BoxData[
    \({{p1\ x - p0\ y + \((p0 + p1)\)\ z, p0\ x + \((p0 + p1)\)\ y + p1\ z}, {
        q1\ x - q0\ y + \((q0 + q1)\)\ z, q0\ x + \((q0 + q1)\)\ y - q1\ z}}
      \)], "Output"]
}, Open  ]],

Cell[BoxData[
    \( (*\ Det[a] = 1\ solution\ for\ q1*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[p1\ q0 - p0\ q1 == 1, q1]\)], "Input"],

Cell[BoxData[
    \({{q1 \[Rule] \(-\(\(1 - p1\ q0\)\/p0\)\)}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(m1 = 
      Simplify[a1 . g] /. {p1\ q0 - p0\ q1 -> 1, 
          \(-p1\)\ q0 + p0\ q1 -> \(-1\), q1 -> \(-\(\(1 - p1\ q0\)\/p0\)\)}
          \)], "Input"],

Cell[BoxData[
    \({{x + z, 
        \(-\((1 - p1\ q0)\)\)\ \((\(-y\) + z)\) + p1\ q0\ \((y + z)\)}, {
        \(-y\) + z, 
        \((1 - p1\ q0)\)\ \((x + y)\) + 
          p1\ \((q0\ \((x + y)\) + \(2\ \((1 - p1\ q0)\)\ z\)\/p0)\)}}\)], 
  "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[Det[m1]]\)], "Input"],

Cell[BoxData[
    \(\(\(-2\)\ p1\ \((\(-1\) + p1\ q0)\)\ z\ \((x + z)\) + 
        p0\ \((x\^2 + y\^2 - y\ z + 2\ p1\ q0\ y\ z + z\^2 - 
              2\ p1\ q0\ z\^2 + x\ \((y + z)\))\)\)\/p0\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[
      \(1\/p0\) 
            \((\(-2\)\ p1\ \((\(-1\) + p1\ q0)\)\ z\ \((x + z)\) + 
                p0\ \((x\^2 + y\^2 - y\ z + 2\ p1\ q0\ y\ z + z\^2 - 
                      2\ p1\ q0\ z\^2 + x\ \((y + z)\))\))\) - 1 == 0, p1]
      \)], "Input"],

Cell[BoxData[
    \({{p1 \[Rule] 
          \(\(-z\)\ \((\(-2\)\ x - 2\ p0\ q0\ y - 2\ z + 2\ p0\ q0\ z)\) - 
              \@\(z\^2\ 
                    \((\(-2\)\ x - 2\ p0\ q0\ y - 2\ z + 2\ p0\ q0\ z)\)\^2 - 
                  4\ p0\ q0\ z\ \((2\ x + 2\ z)\)\ 
                    \((1 - x\^2 - x\ y - y\^2 - x\ z + y\ z - z\^2)\)\)\)\/\(2
              \ q0\ z\ \((2\ x + 2\ z)\)\)}, {
        p1 \[Rule] 
          \(\(-z\)\ \((\(-2\)\ x - 2\ p0\ q0\ y - 2\ z + 2\ p0\ q0\ z)\) + 
              \@\(z\^2\ 
                    \((\(-2\)\ x - 2\ p0\ q0\ y - 2\ z + 2\ p0\ q0\ z)\)\^2 - 
                  4\ p0\ q0\ z\ \((2\ x + 2\ z)\)\ 
                    \((1 - x\^2 - x\ y - y\^2 - x\ z + y\ z - z\^2)\)\)\)\/\(2
              \ q0\ z\ \((2\ x + 2\ z)\)\)}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(m2 = 
      Simplify[g . a1] /. {p1\ q0 - p0\ q1 -> 1, 
          \(-p1\)\ q0 + p0\ q1 -> \(-1\), q1 -> \(-\(\(1 - p1\ q0\)\/p0\)\)}
          \)], "Input"],

Cell[BoxData[
    \({{\(-p0\)\ 
            \((\(-\(\(\((1 - p1\ q0)\)\ \((x + y)\)\)\/p0\)\) + 
                q0\ \((y - z)\))\) - 
          p1\ \((\(-q0\)\ \((x + z)\) - \(\((1 - p1\ q0)\)\ \((y + z)\)\)\/p0)
              \), p0\^2\ \((y - z)\) + p0\ p1\ \((y - z)\) + 
          p1\^2\ \((y + z)\)}, {
        \(-\((q0\^2 - \(q0\ \((1 - p1\ q0)\)\)\/p0 + 
                \((1 - p1\ q0)\)\^2\/p0\^2)\)\)\ \((y - z)\), 
        p1\ \((q0\ \((x + y)\) - \(\((1 - p1\ q0)\)\ \((y - z)\)\)\/p0)\) - 
          p0\ \((\(-\(\(\((1 - p1\ q0)\)\ \((x + z)\)\)\/p0\)\) + 
                q0\ \((\(-y\) + z)\))\)}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[Det[m2]]\)], "Input"],

Cell[BoxData[
    \(\(\(-2\)\ p1\ \((\(-1\) + p1\ q0)\)\ z\ \((x + z)\) + 
        p0\ \((x\^2 + y\^2 - y\ z + 2\ p1\ q0\ y\ z + z\^2 - 
              2\ p1\ q0\ z\^2 + x\ \((y + z)\))\)\)\/p0\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[
      \(1\/p0\) 
            \((\(-2\)\ p1\ \((\(-1\) + p1\ q0)\)\ z\ \((x + z)\) + 
                p0\ \((x\^2 + y\^2 - y\ z + 2\ p1\ q0\ y\ z + z\^2 - 
                      2\ p1\ q0\ z\^2 + x\ \((y + z)\))\))\) - 1 == 0, p1]
      \)], "Input"],

Cell[BoxData[
    \({{p1 \[Rule] 
          \(\(-z\)\ \((\(-2\)\ x - 2\ p0\ q0\ y - 2\ z + 2\ p0\ q0\ z)\) - 
              \@\(z\^2\ 
                    \((\(-2\)\ x - 2\ p0\ q0\ y - 2\ z + 2\ p0\ q0\ z)\)\^2 - 
                  4\ p0\ q0\ z\ \((2\ x + 2\ z)\)\ 
                    \((1 - x\^2 - x\ y - y\^2 - x\ z + y\ z - z\^2)\)\)\)\/\(2
              \ q0\ z\ \((2\ x + 2\ z)\)\)}, {
        p1 \[Rule] 
          \(\(-z\)\ \((\(-2\)\ x - 2\ p0\ q0\ y - 2\ z + 2\ p0\ q0\ z)\) + 
              \@\(z\^2\ 
                    \((\(-2\)\ x - 2\ p0\ q0\ y - 2\ z + 2\ p0\ q0\ z)\)\^2 - 
                  4\ p0\ q0\ z\ \((2\ x + 2\ z)\)\ 
                    \((1 - x\^2 - x\ y - y\^2 - x\ z + y\ z - z\^2)\)\)\)\/\(2
              \ q0\ z\ \((2\ x + 2\ z)\)\)}}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\( (*\ 
      p1\ solutions\ show\ the\ group\ behaves\ as\ Abelian\ 
            \((a^\(-1\))\) . g - g . \((a^\(-1\))\) == 0*) \ \)\)], "Input"],

Cell[BoxData[
    \( (*\ 
      RATIONAL_Implicit = 
        \(1\/p0\) 
              \((\(-2\)\ p1\ \((\(-1\) + p1\ q0)\)\ z\ \((x + z)\) + 
                  p0\ \((x\^2 + y\^2 - y\ z + 2\ p1\ q0\ y\ z + z\^2 - 
                        2\ p1\ q0\ z\^2 + x\ \((y + z)\))\))\) - 1 == 0\ *) 
      \)], "Input"],

Cell[BoxData[
    \(\( (*\ for\ p0 = 1, p1 = 2, 
      q0 = 3\ this\ results\ in\ a\ Catenoid\ like\ 3  d\ surface*) \n (*\ 
      such\ a\ surface\ as\ seen\ in\ a\ Bryant\ cousin\ Minimal\ surface*) \ 
    \)\)], "Input"]
},
FrontEndVersion->"Macintosh 3.0",
ScreenRectangle->{{0, 1920}, {0, 1060}},
ScreenStyleEnvironment->"Condensed",
WindowSize->{1196, 921},
WindowMargins->{{172, Automatic}, {Automatic, 14}},
PrintingCopies->1,
PrintingPageRange->{1, Automatic},
MacintoshSystemPageSetup->"\<\
00/0004/0B`000003509H?ocokD"
]


(***********************************************************************
Cached data follows.  If you edit this Notebook file directly, not using
Mathematica, you must remove the line containing CacheID at the top of 
the file.  The cache data will then be recreated when you save this file 
from within Mathematica.
***********************************************************************)

(*CellTagsOutline
CellTagsIndex->{}
*)

(*CellTagsIndex
CellTagsIndex->{}
*)

(*NotebookFileOutline
Notebook[{
Cell[1709, 49, 255, 6, 18, "Input"],
Cell[1967, 57, 97, 2, 18, "Input"],
Cell[2067, 61, 90, 1, 18, "Input"],

Cell[CellGroupData[{
Cell[2182, 66, 57, 1, 18, "Input"],
Cell[2242, 69, 54, 1, 18, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[2333, 75, 77, 1, 18, "Input"],
Cell[2413, 78, 74, 1, 18, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[2524, 84, 72, 1, 18, "Input"],
Cell[2599, 87, 69, 1, 18, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[2705, 93, 60, 1, 18, "Input"],
Cell[2768, 96, 168, 3, 34, "Output"]
}, Open  ]],
Cell[2951, 102, 88, 1, 18, "Input"],

Cell[CellGroupData[{
Cell[3064, 107, 52, 1, 18, "Input"],
Cell[3119, 110, 190, 3, 18, "Output"]
}, Open  ]],
Cell[3324, 116, 70, 1, 18, "Input"],

Cell[CellGroupData[{
Cell[3419, 121, 64, 1, 18, "Input"],
Cell[3486, 124, 76, 1, 34, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[3599, 130, 176, 4, 34, "Input"],
Cell[3778, 136, 253, 6, 34, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[4068, 147, 50, 1, 18, "Input"],
Cell[4121, 150, 206, 3, 36, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[4364, 158, 276, 6, 34, "Input"],
Cell[4643, 166, 781, 14, 67, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[5461, 185, 176, 4, 34, "Input"],
Cell[5640, 191, 625, 11, 69, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[6302, 207, 50, 1, 18, "Input"],
Cell[6355, 210, 206, 3, 36, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[6598, 218, 276, 6, 34, "Input"],
Cell[6877, 226, 781, 14, 67, "Output"]
}, Open  ]],
Cell[7673, 243, 166, 3, 18, "Input"],
Cell[7842, 248, 310, 7, 26, "Input"],
Cell[8155, 257, 223, 4, 32, "Input"]
}
]
*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)

 





-- 
Respectfully, Roger L. Bagula
tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/ 


  • Prev by Date: a noise with a better histogram
  • Next by Date: Sum of list elements
  • Previous by thread: Re: a noise with a better histogram
  • Next by thread: Sum of list elements