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Pascalloid polyhedral geometries in 3D (4D) can be represented using this Pascalloid Calculator, though full expanded algebra terms are not included in the number orbs, they could be. This project is posted and maintained at SourceForge.net 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.

Recent updates include 'Radius Coloring', a 'Size Adjust', and finally accurate Icosal and Dodecahedral Pascalloids, plus User-Defined Pascalloids can finally be created!  The internal contextual help text was also revised and updated.

Version 9.1 still has the polar (+/-) mode, but the 216-var Deltoidal Icositetrahedron and Rhombic Cuboctahedron have 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 algorythm (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 / all platforms version, we get to run in 64 bit. I am not a licensed Apple Store developer.  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 number base (radix) is the most awesome accomplishment.

Mobile device users:  please be careful with your device and stay to low N values.  This program was built with Adobe Flash 8, which can still run in Safari with the correct security settings, will easily run in Chrome on any computer, can run using the Puffin browser on an Android device, and can run using the Photon browser on any Apple touch-screen, via this link:

Pascalloid Calculator 9.1 for Browser

Also, ready to download for later convenience:

Pascalloid Calculator 9.1 for Mac Intel x86 and All Platforms

Pascalloid Calculator 9.1 for Win/PC, (.exe)

Pascalloid Calc 9.1 for Mac OS, PPC


We also have a stand-alone .exe that amazingly works on all versions of Windows.  No version of this program will connect back to the internet for any reason, it is for the sake of knowledge. It 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 users, 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 program now starts in small-screen mode, and pressing the 'w' key or clicking the grey scree-resize button will quickly activate full-screen.

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.  The help window is touch-screen scrollable with only a minor defect; it will copy help text into memory automatically when clicked.

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.

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.

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Thomas.Chenhall@gmail.com