Kirjoitettua

2025-09-09

2025-09-09_20:46:30 PSL and Polyhedrons

PSL, Poutvaara Science Laboratory, is a new branch started from the Design Bureau Poutvaara. Similar as Poutvaara Space Agency, also PSL is focused on modelling and visualization, but more in geometry and engineering design.

After PSA was investigated the spherical shape of Earth and the gravitational related orbits, PSL has been investigating perfect geometrical forms, more abstract and mathematical in nature. The starting project of the PSL is about Polyhedrons, geometrical shapes, which can also be found in nature. Polyhedrons are regular geometry-defined forms, from which there can be made objects. As focus there are the most universal Platonic Polyhedrons (5) and also as famous Archimedean Polyhedrons (13).

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

Platonic solids are regular polyhedrons, which have been known from ancient times. They are known after Plato’s writing, that connected those in cosmic and alchemic explanations. Their mathematical characteristics and complex simplicity have fascinated people for thousands of years.

Archimedean solids are also regular polyhedrons, that have uniform character and some of them are connected to Platonic solids. Some of them are very complex to calculate and model. Also, these have fascinated people from the ancient times and even in present times these can be found in many aspects of human activities, from constructions such as weather and airport radar covers literary in almost every country and even in majority of every soccer balls.

Other famous solids are Catalan solids (13), Deltahedrons (8), Johnson solids (92), Goldberg polyhedrons (mosaic system) and Kepler – Poinsot polyhedron (4). In architecture the best known are geodesic dome concept, most popular in the 1960s, mainly based on now expired patented innovations. As most solids are already known – architects, designers and scientists have been focusing on inventing irregular and nature related forms. Self I might have interest also in some of these newer solids in some other occasion. Most of the regular ones are similar to model as the Platonic and Archimedean. The most complex class is the Goldberg mosaic system that has a different type of modelling system and more limited use. The best known and the most durable solid forms are the pyramids. Although they are considered to be simple, actually almost all of them have actually been different even since the ancient Egyptian times. The more complex the geometric system is, the more difficult it is to apply to material form. In practice this means compromises and simplifying in the realization process.

PSL has better and more accurate 3D-models about the Platonic and Archimedean solids than Wikipedia, but the main reference has actually been the Wiki-pages. Those do not include any final calculations or advices, but they act as confirmation for the theory, which can also be checked from the final model. I would emphaticize that the requirements are university level mathematics and advanced CAD skills. The Wiki-pages do not include ready solutions, so if anyone is interested of the subject, so rather than giving a ready 100 pages calculations as a guide, I would suggest one to get the skills needed, they could be useful also in other fields. Calculations take at least a week and modelling about one month – if one has good calculation routines and professional modelling skills.

The edges of polyhedrons are precise up to calculator devices and programs accuracy – practically this means basic unit precision with either 1:100 000 000 or 1:1 000 000 000. This would sufficient for any material realization. If anyone is interested in the scientific side of the subject, the Polyhedrons can be presented as calculations with models, but only in scientific use. In practice the skills used and any potential use are financially valuable, so the models or the theory is not to be distributed without compensation. My calculations and modellings have taken about a month so they include use of resources and working hours. For the use of science PSL offers knowledge, but only as a main author of the project.

The most complex mathematical formula used in all Platonic and Archimedean solids is the following:

It might look complicated, but I see it as elementary college level mathematics, that requires routine calculation skills. If one sees it difficult, one probably should revise the mathematical skills and routines. Modelling with it is a little bit more complicated, but with my 6000+ hours CAD-use background I found it simple. My own model about the project as a whole fitted in one 0.7 Mb central file with 1.4 Mb of 27 final solids as xrefs. Its editing time was 100 hours. Practically one of the most requiring tasks was to calculate all 2200 vertices and modelling them. Also defining 1000 coordinate systems required patience. Self the solid modelling was very simple, but in case of a small previously unnoticed mistake it was stressful.

My project models are precise models including vertice points, edges and solids. The 3D-AutoCAD models are with either 1:100 000 000 or 1:1 000 000 000-unit accuracy. My model includes only necessary and precise information, but it consists of 7000 lines, 1000 texts, 1000 coordinate systems, 2200 points and 500 layers. Mathematical calculations are 100 pages and they are correct, have been checked from the model and the formulas related use wiki-pages as reference. Self the whole project is scientifically valid. The wiki-pages, however, do not include any precise advices or final calculations. Besides, the wiki-pages includes only semi-professional models. PSL has commercially useable professional level models.

When one makes the calculations and modellings, it is very important that everything is precise. There is no easier way of doing this professionally. With 9-digit unit calculation precision with calculator and 8-digit unit precision with AutoCAD one can reach only 0.000 000 01-digit modelling unit precision and for AutoCAD there is no more accurate precision available. It would however, in realization of a 300 m dome or a spherical structure be in a virtual model as 0.03 mm or 0.003 mm. With 300 meters dome that in reality of realization of a steel structure assembly could be something like 0.25 mm. That could proof, that my solids fit for modelling of any type of physical structure.

Similarly, as with Poutvaara Globe, I would offer these Platonic and Archimedean solids in different size classes. Examples are freely scalable and can be for example:

Human size models: 1 cm, 2 cm, 3 cm, 5 cm, 10 cm, 20 cm, 50 cm, 70 cm, 1 m, 1.5 m and 2 m.

Sculpture and small building size models: 3 m, 6 m, 12 m, 20 m and 30 m.

Large building size models: 40 m, 50 m, 80 m, 100 m, 120 m, 150 m, 180 m, 200 m, 250 m, 300 m and up to about 350 m.

The polyhedrons can be used in several ways: as firstly mentioned, the scientific use. More related to my company, we offer artistic use, design use, architectural use and engineering use applications of polyhedrons. We do not offer only modelling. Similarly, as with Poutvaara Globe model, also these solids have an abstract function to act as pure virtual mathematical models. These could be used in science, for example in models of natural structures. Similarly, they can be realized as 3D-objects. PSL offers these models for use as leading architectural concept designer, artist, scientist, engineer or designer, but for more complicated tasks the rights can be divided. PSL reserves all original rights for the model data.

The Platonic and Archimedean Polyhedrons do not actually belong to either person, Platon or Archimedes, but are actually ancient Greek, Pythagorean and Egyptian inventions and grouped or classified after the two scientists. They are at least about 2200-2500 years old. Self I have not made any new scientific innovations about the polyhedrons, but I have made own calculations and own modellings, that no other person has presented in internet. Moreover, I am able to use my own modellings in science, art, design, architecture and engineering. Innovating new non-regular solid forms in architecture is quite popular and relatively easy to realize as most of the uniform polyhedrons are already known. There is still space to innovate and find, but before that one should know what is already known.

In Wikipedia the form and similar are in category “Polyhedron”. Polyhedron is a multi-use form. Its basics are in geometry and mathematics, but its use is related to all aspects of life: arts, architecture, engineering, military use, biology, chemistry, physics, mapmaking – practically all modern time humans have benefitted in some form of the concept polyhedron – either they have known about it or not. Maybe best illustrative form is that aviation traffic control and weather forecasting are supported by information produced by radars using polyhedron form as cover – everyone, who have travelled with an airplane or seen a weather forecast have been using information made with a polyhedron. Polyhedrons are in character multi-use forms, working both in military and civilian world, interconnected in arts, design, engineering, architecture, science etc. in various interconnected ways. PSL is simply only giving its contribution to its 2200 years old concept. Primary PSL is able to use contemporary polyhedron models.

Polyhedrons are not particularly a 20th century idea – the origins of the idea were created in the ancient Greece and it was rediscovered in several periods such as in the Renaissance era. In the 20th century it was widely adapted by engineers and architects using modular construction techniques and as its one leading spokesperson in public media was selected as Buckminster Fuller. The invention would have been as popular without him, but Fuller was able to bring new aesthetics for the subject. Later as the technical limitations of the form became obvious, there was new tendency of random non-geometrical forms and imitations of nature, such as with projects derived from bubble surfaces. Though not being any more popular in modern architecture, the polyhedrons are very popular in technical structures such as radar covers because of simple construction, cost cutting and reliability. The dual use of polyhedrons is widely known and there is nothing to do about it. Mastering using of polyhedrons is possible even without a university mathematics degree and modelling it for practical use is possible without an engineering degree.

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.