Pascalloid polyhedra geometries in 3D (4D) can be represented using this Pascalloid Calculator, though full expanded algebra terms are not yet included in the number orbs, they could be. This project is posted and maintained at SourceForge.com as “Pascal's Triangle X Plato's Polyhedra” a mathematics project under GPL license. It exposes only the first stage of an immense well of geometric form, and can get a little overwhelming. As we attempt to see only the finite, after about every two minutes of solid computation, the “do you want to abort script?” warning will appear, and to provide a sense of suspense that it may or may not work, or even be working at all, but the program actually works quite well at this stage, and is challenging / confusing.
Version 8.9 still has the polar (+/-) mode, but the 216-var Deltoidal Icositetrahedron has been switched out. Up and down arrow keys now control 'Focal Length' of the view, or apparent nearness of the geometry, and the text size in Help (?) mode and in Data Only mode. The geometry of the "Tri-Decahedron" was added at some point, as a compact useful structure. A 5x5x5 "Tesseract" geometry was also added, providing 3D expansion of the 2D Fruit of Life alg (see Pascal's Canvas). Color set “thermostats” [a], [s], [d], [f] and [g] are wilder now. Constraint parameters in upper-right text boxes round themselves off once return is pressed; to nearest displayable shelf-values, automatically. Incrementing the constraint parameters jumps from one displayable shelf-value to the next shelf-value. In this updated Mac Intel version, we get to run in 64 bit. I am not a licensed Apple Store developer, but I swear this is entire project is for educational purposes and not intended to make money. Those with Mathematica, Maple, Sage or another math program may do the algebraic Expansions and prove the coefficients. Side note: the Horizontal Addition base (radix) is the most awesome accomplishment.
This program utilizes
Adobe Flash 8, which can still run in Safari with the correct security
settings, will easily run in Chrome on any computer, and can run using
Photon on any Apple touch-screen
Ready to download for later convenience:
Pascalloid Calculator 8.9 for Mac OS, Intel chip
Pascalloid Calc 8.9 for Mac OS, Motorola chip
Pascalloid Calculator 8.9 for Win/PC, (.exe)
8.9 for any platform (.swf)
We also have a stand-alone .exe that amazingly works on Windows XP, Vista, Windows 7 and Windows 8. No version of this program will connect back to the internet for any reason, it is for the sake of knowledge. It generates its own structures like a Mandelbrot zoom program, in that data is not pre-stored and exhibits fractal / self-similar patterns (especially once the 'Horizontal Addition (base Radix)' function is applied).
Esc will now automatically quit the program, but it won't start full-screen so the operating program window is still manageable.
For Windows, we have our decent stand alone .exe.
Again, if you're going to try and test your maximum reachable
N-values, please do not leave it in full screen, or could have a
hard time force-quitting it.
The primary results of significance are encoded in a new geometry notation system that can be found in the '?' button under the “Hot Numbers” info button. Because the Deltoidalicositetrahedron is now correct, it is extremely hard to get to N=2, and I have not ferreted out many geometric transmutations, whereas with the simpler version having no equation I had found plenty of interesting geometric transmutations. The help window is touch-screen scrollable with only a minor defect; it will copy into memory automatically when clicked so that I can easily repair the strings that store the help text when they need spell checking.
The inspiration for this project was not kindled from nowhere. I was fascinated by Buckminster Fuller's Concentric Hierarchy of Shape – a really good website using animated .gif's to teach geometry in a new way. Pascal's triangle I learned from school, not a web-page, but here's a decent one.
Also, the best overview of shapes and their understanding, free from clutter, is Polyhedra.org– an excellent and computationally economical way to see new potential Pascalloid target shapes (with more than just corners), though each one would prove more work than any one person could ever complete.
There is a very good Wikipedia entry for 'Pascal's Pyramid' (parallels the Pascalloid Triangle). There is yet none for 'Pascal's Hyper-Tetrahedron' (what a stark novelty), and also no entry for Pascalloid(s) yet. There is a very advanced entry on Wikipedia for the 'Normal Curve', that may explain another difficult goal.