After 400 years, mathematicians find a new class of solid shapes

After 400 years, mathematicians find a new class of solid shapes
(source: theconversation.com)

The work of the Greek polymath Plato has kept millions of people busy for millennia. A few among them have been mathematicians who have obsessed about Platonic solids, a class of geometric forms that are highly regular and are commonly found in nature.

Since Plato’s work, two other classes of equilateral convex polyhedra, as the collective of these shapes are called, have been found: Archimedean solids (including truncated icosahedron) and Kepler solids (including rhombic polyhedra). Nearly 400 years after the last class was described, researchers claim that they may have now invented a new, fourth class, which they call Goldberg polyhedra. Also, they believe that their rules show that an infinite number of such classes could exist.

Platonic solids, Archimedean solids, Kepler solids, and now, a new class called "Goldberg solids".

Platonic solids:



Platonic solids occur in nature, example iron sulfide:



Example of a Kepler solid;



The new class came as researchers (Stan Schein, James Gayed, UCLA) looking into the human eye took note of the polyhedra they were finding, and related it back to a mathematician named Michael Goldberg, who "described a set of new shapes, which have been named after him, as Goldberg polyhedra."

They described new polyhedra based on Goldberg's research, but went way beyond it; example, a "blown up dodecahedron"



They believe it will play a vital role in virus research; building research and design among other uses.

Examples of shapes possible with 'Goldberg solids';



Interesting article at the link above, just another fascinating aspect of nature to mull over.

I gotta get some of those cubes and make a set of dice out of them. It would be a good conversation piece....lost amongst all the junk in our house.

rickymouse
I gotta get some of those cubes and make a set of dice out of them. It would be a good conversation piece....lost amongst all the junk in our house.

Too bad Gary Gygax didn't live to see the "infinite sided dice".

This article goes a bit more into the scientific application of 'Goldberg polyhedra';

Goldberg variations: New shapes for molecular cages
(source: sciencenews.org)


Caption: SHAPE SHIFTER A new type of molecular cage has all sides of equal length and flat faces that are all pentagons or hexagons (left). The hexagons have angles that vary from 104 to 142 degrees. A fullerene with the same number of vertices and the same pattern of faces (right) has hexagons with angles that vary in a much narrower range. This forces the hexagonal faces to warp and the cage to assume a pointy shape.


Caption: Stan Schein (left) and James Gayed are not mathematicians, but they are polyhedron lovers.

reply to post by Blackmarketeer


Not sure if I'm understanding this. Taking a shape and compounding it into an identical shape composed of numerous miniature models of that shape constitutes as a new shape?

reply to post by Blackmarketeer


Those look like the balls our wolf likes to chase. Kick them off the porch into the yard, or three or four of them at a time around the yard, and he's in his own form of heaven.



So he discovered the Goldberg variations long ago. Nice thread, but since I'm math illiterate I'll watch and learn.

rickymouse
I gotta get some of those cubes and make a set of dice out of them. It would be a good conversation piece....lost amongst all the junk in our house.

Yes - new maneuvers in Dungeons and Dragons are already stirring!

Please excuse my ignorance, but I have to ask. What is the difference between these and geodesic domes? I don't see a difference, but I'm not that bright either. Could someone please enlighten me?

Buckminster Fuller

AfterInfinity
reply to post by Blackmarketeer

 

Not sure if I'm understanding this. Taking a shape and compounding it into an identical shape composed of numerous miniature models of that shape constitutes as a new shape?

I believe it has to do with the mathematics at the vertices where each flat face connects to it's neighbor. Polyhedron is Greek for 'many faces'. Each of the so-far known solids (Platonic, Archimedean,etc.) has a different mathematical relationship at these vertices.

reply to post by Blackmarketeer


That is very cool and since every shape has a vibration and every vibration has a color and sound ,that means each of these new shapes, has a new vibration and a sound that corresponds to it ,that we may have never heard before or can't because of its vibrational level,they did an experiment on Platonic solids and the geometric shapes at rossyln chapel to discover this and according to their research this is one of the secrets that ancient mystery cults already knew about long ago ,I wonder if the ancients knew about this and we are just rediscovering it now.

reply to post by Blackmarketeer


It sounds to me like they made a painting and called it a new color.

Oh, wait...is this news because they expanded the number of sides? Like they graduated from the pentagon to a hexagonal polyhedron? I didnt know it was that hard.

A six-sided base for a Buckminsterfullerene? Odd I wouldn't have considered that is a new shape...but what do I know?

Surely it won't take 400 more years to add the sides to 7 or 8? LOL

reply to post by Aleister


I am absolutely horrible at math as well ,but if you do some quick research on Platonic solids,and what is called sacred geometry it is a bit easier to understand ,you begin to see the patterns in nature and how math and numbers rule the entire universe and how everything in nature is connected by math ,it's such a fascinating subject that I respect and awe over even tho I hate math,the golden ratio or pi is just one great example of how math can be seen in nature on the inside of a seashell or the way a galaxy spirals.i don't fully understand every aspect of the subject as well as I'd like to and thats because i suck at math ,but knowledge is power and there is always time to learn something new ,like I said do some reading on sacred geometry ,platonic solids and basic math of the universe and you will have a slightly better grasp or idea of the subject.