"OPTi" is a Macintosh application program which visualizes quasi-conformal deformations of the once-punctured-torus groups. It interactively draws the isometric circles, the Ford region, the limit set, etc.

Corrected the link to OPTi 3.61. Sorry for the inconvenience. (2011.6.7)

There is nothing really new, but I have packaged OPTi 3.61 with 3.60 documents in zip format, rather than sit format. 2010.12.4.

Now I provide the source codes of OPTi for those who are interested in the inner workings of OPTi. Modification of the program is welcomed.

Also, there has been a request for my old program called "CyclicGroup", and I am making it together with its source codes available for download.

I have uploaded a few of my preprints related to OPTi. See below in the download section.

OPTi goes 3D!! Now OPTi shows the Ford region of a given group in a realistic 3D view. Just choose "Show 3D View" from the "Group" menu. The OpenGL graphics library, which is needed in order to use the menu, is included in Mac OS X as well as in recent versions of Mac OS 9. But if you are using older versions of Mac OS, you may need to download and install appropriate versions of CarbonLib and/or OpenGLLibrary from the Apple website.

Also OPTi 3.60 now uses more feasible colors for the parameter space of the Maskit slice. The randomly looking colors shown in the Maskit's parameter space by OPTi 3.50 were due to numerical errors. Cusps are very unstable in the sense that infinitely many groups having different Ford region types reside near each cusp. Thus a very small numerical error could cause a group to look having a totally different Ford region from the original. Therefore, the pictures of Maskit slice obtained by OPTi 3.50 and by OPTi 3.60 are both correct in the mathematical sense, but of course we prefer the latter.

**OPTi 3.61** --- OPTi 3.60 User's Manual, and some
sample files are archived as a zip file. The program runs under
Mac OS X (using Rosetta for Intel Macs) as well as Mac OS 9.

**OPTi
3.61 source files** --- CodeWarrior Pro 8.3 project and source
files, archived in zip format. Modification of the program is welcomed,
if anybody is interested.

"OPTi 3.60" program, OPTi User's Manual, and some sample files are archived as a BinHexed StuffIt file. The program runs under Mac OS X as well as Mac OS 9.

You will need "StuffIt Expander" (freeware) to expand the downloaded file.

**OPTi
3.60** --- runs on Mac OS X as well as Mac OS 9. (357KB)

**OPTi 3.50** --- runs on Mac OS X as well as Mac OS 9. (308KB)

If you use Mac OS 8 or older, you may need to use an older version of OPTi:

**OPTi
3.30** --- runs on PPC macs only. (461KB)

**OPTi
3.21** --- runs on both PPC and 68K macs. (771KB)

**CyclicGroup** --- Classic application, and its source files.

Here are some documents about OPTi in PDF format.

- OPTi's algorithm for discreteness determination (Japanese) --- explains the mathematical background of OPTi. The printed version of this is also available in RIMS Kokyuroku 1270 "Hyperbolic Spaces and Discrete Groups II" (2002), 67-76.
- OPTi's algorithm for discreteness determination --- This is essentially an English translation of the above.
- OPTi's algorithm for drawing the limit set (Japanese) --- This is a note for my talk in "Topology and Computer" held at Nara Women's University, November 29 - December 1, 2001. It contains a short informal description of how OPTi draws the limit set, together with an introduction to programming for mathematicians.
- OPTi's algorithm for drawing the limit set --- an English translation of the above.

"OPTi 3.60" is a freeware program. You may freely copy and distribute "OPTi 3.60".

If you use pictures produced by this program for your publication, I would appreciate it very much if you send me a copy of the pulication (whole or part) containing the picture, or at least tell me where I can find it.

Every program of this complexity contains a bug or two. If you find any, please report it by sending email to wada@ist.osaka-u.ac.jp. It would greatly help me improve the program.

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