Kirjoitettua

2025-09-28

2025-09-28_16:18:30 PSL and more Polyhedrons

PSL, Poutvaara Science Laboratory, has advanced in some of the newer polyhedrons, namely modelling the concepts of 1915 (Rausenberger) and 1964 (Bernal) defined deltahedron solids. Some of them are even original Platonic solids such as Regular tetrahedron, Regular octahedron and Regular icosahedron, but have received new companions. The Catalan solids after a list by Eugène Catalan were completed in 1865. In addition to the ancient solids, PSL has now realized them in digital format using contemporary technology and advanced mathematics.

In the previous writing there was about the Platonic and Archimedean solids modelling. Those were ancient inventions grouped after the two historical scientists. The Deltahedrons and the Catalan solids included more irregular forms with thus some extra challenges.

For the basic concepts I would thank Wikipedia edited in English, German, Russian, French and Spanish, Wolfram MathWorld, Dmccooey, Polytype Wiki, Rechneronline, Google Ai and Wolfram Alpha. The most useful sources were Wiki-pages and Wolfram -pages and the two polyhedron calculators. Apart from Google Ai, these do not include starter level -information or exact guides and the most complicated tasks can be solved only by the scientist alone. Google Ai don’t provide much anything for an expert, but it helps with finding internet sites. My own modelling experience is working only for CAD-experts with academic background. It is, however, difficult even for most architects, as most of them are not specialized in modelling. In YouTube, for example, one can find videos about modelling polyhedrons with computer, but I found most of them irrelevant for me, because they included too many shortcuts.

All volumes presented in one picture, to see more closely, visit PSL.

The Deltahedron solids are thus simple, that their faces are equilateral triangles. The Catalan solids are more complex, as they are less regular. The most complex of them is the last, the Pentagonal hexecontahedron, from which there is only with about 0.00015 -unit precision information available in Wikipedia. After several days of work, I got it right and now I have also of all Deltahedron and Catalan solids a model with 0.000 000 01 or 0.000 000 001-precision. This unit precision counts, as the radius is usually 1 unit.

So far, the Catalan solids have been in some contemporary advanced use, such as pentagonal hexecontahedron in the Amazon Spheres architecture project, Seattle, and disdyakis triacontahedron in some technical aspects of grid system in satellite imagery and creating of geodesic spheres. All of them have also various artistic and design use.

To illustrate the question of accuracy, I would take into examination about Pentagonal hexecontahedron their use in the Amazon Spheres in Seattle. In the realized project there are 25, 30 and 40 m diameter steel- and glass spheres built of prefabricated units. With Wikipedia accuracy the 40 m sphere would have an error of 6 mm, which is not tolerable in steel and glass assembly. I invented my own model with 0.000 000 01 accuracy, because there was no precise information available in internet or any Wiki-pages. My own model accuracy projected on error marginal of 40 m sphere has an error marginal of only 0.0004 mm and for a 350 m sphere 0.0035 mm. That would fit for any practical materialization.

View of the stacked solids made from formula by Wikipedia with 0.00015 -unit precision in red and my own with 0.000 000 01 accuracy in gray. The error might seem to be small, but in architecture size object realization it counts.

View of disdyakis triacontahedron projected into size of Earth. The form is spherical, and by using great circles, it can be transformed into 15 Earth -radius circles. This form of coordinate system is evenly divided unlike the current Earth map grid. The system could be used in separate contexts, such as for satellite mapping use. The system has no potential to replace the common mapping system, but it can be used by specialists.

The predominant mapping system with 5 degree -interval grid compared with uniform size grid system from great circles derived from the form of disdyakis triacontahedron. There are no common standards developed, such as naming and division of elements. Replacing the GPS-related mapping system seems not likely and even the Military Grid Reference system relies on the Universal Transverse Mercator coordinate system. Tasks by satellite mapping are different.

The variety of using polyhedrons from PSL created models are following:

1. Scientific use, design of uniform objects
2. Theoretical scientific use, for example mathematics and geometry
3. Architecture, Engineering and Design
4. Artistic use

We can provide practical and scientifically reliable solutions for a large variety of uses. We continue Research and Development work similarly as practical applications.

Now after modelling 39 types of polyhedrons – Platonic, Archimedean, Deltahedrons and Catalans, the short journey going through the most important types is done. Crystallography and Johnson solids are not covered in modelling required details in Wikipedia so that would not be the next interest. Instead, focusing in the actual applications of the modelled polyhedrons has been in mind.