After 400 Years, Mathematicians discover a new class of Shape

Knowing me, as soon as I saw an article about a new kind of geometry I was rather giddy! Essentially, a new “Class” of geometry was recently discovered after 400 years of research, building on our knowing of the ancient and rudimentary geometric structures first discussed by Plato several thousand years ago. 


The Trinity – Not a “Platonic” solid, but certainly an important one at that.

Upon reading the article below, at first I was a little confused. Primarily because sometimes when Mathematicians find stuff out, it’s very left-brained, very mathematical, and often leaves me pondering what the shape truly means, and how it relates to myself and the world around me in a more female, emotional, thought-based way.

What implications does this geometry have on my life?

You might remember the recent Spirit Science video about the trinity –  fun fact about that – it’s technically not a shape that can exist with straight lines, thus it cannot be a “platonic” solid. It certainly is a solid though, and as we theorized in the video – is it the geometry of a straight line itself?

Regardless of that, anything to do with geometry is incredibly exciting to me, especially when it’s relateable to our everyday world. Although this new discovery may not have specific relatable attributes to the way we live our lives in an immediate sense, it does relate a lot to the molecular structures of everything, including (as noted at the bottom of the article), viruses and diseases.

Perhaps through awareness of this structure, we can more accurately match the vibration of viruses, and cure more diseases and create more healing on this planet.

(Now, it’s probably more easier said than done, if only because… well, you know :P) 

Now, as you’re looking at the pictures below, notice what these geometries are made of, and why they are important. When you break it down, it’s individual shapes composed of 5 and 6 sided shapes (Pentagons & Hexagons), arranged in a rather elegant pattern…

It’s much more complex than the platonic basics, of course, but nonetheless an important building block, and piece to understanding our world.

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

By Akshat Rathi, The Conversation

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 regular convex polyhedra, as the collective of these shapes are called, have been found: Archimedian solids and Kepler solids. 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 love for geometry

Regular convex polyhedra need to have certain characteristics. First, each of the sides of the polyhedra needs to be of the same length. Second, the shape must be completely solid: that is, it must have a well-defined inside and outside that is separated by the shape itself. Third, any point on a line that connects two points in a shape must never fall outside the shape.

Platonic solids, the first class of such shapes, are well known. They consist of five different shapes: tetrahedron, cube, octahedron, dodecahedron and icosahedron. They have four, six, eight, twelve and twenty faces, respectively.

Platonic solids in ascending order of number of faces.

These highly regular structures are commonly found in nature. For instance, the carbon atoms in a diamond are arranged in a tetrahedral shape. Common salt and fool’s gold (iron sulfide) form cubic crystals, and calcium fluoride forms octahedral crystals.

The new discovery comes from researchers who were inspired by finding such interesting polyhedra in their own work that involved the human eye. Stan Schein at the University of California in Los Angeles was studying the retina of the eye when he became interested in the structure of protein called clathrin. Clathrin is involved in moving resources inside and outside cells, and in that process it forms only a handful number of shapes. These shapes intrigued Schein, who ended up coming up with a mathematical explanation for the phenomenon.

Goldberg polyhedron.

During this work, Schein came across the work of 20th century mathematician Michael Goldberg who described a set of new shapes, which have been named after him, as Goldberg polyhedra. The easiest Goldberg polyhedron to imagine looks like a blown-up football, as the shape is made of many pentagons and hexagons connected to each other in a symmetrical manner (see image to the left).

However, Schein believes that Goldberg’s shapes – or cages, as geometers call them – are not polyhedra. “It may be confusing because Goldberg called them polyhedra, a perfectly sensible name to a graph theorist, but to a geometer, polyhedra require planar faces,” Schein said.

Instead, in a new paper in the Proceedings of the National Academy of Sciences, Schein and his colleague James Gayed have described that a fourth class of convex polyhedra, which given Goldberg’s influence they want to call Goldberg polyhedra, even at the cost of confusing others.

Blown up dodecahedron. stblaize

A crude way to describe Schein and Gayed’s work, according to David Craven at the University of Birmingham, “is to take a cube and blow it up like a balloon” – which would make its faces bulge (see image to the right). The point at which the new shapes breaks the third rule – which is, any point on a line that connects two points in that shape falls outside the shape – is what Schein and Gayed care about most.

Craven said, “There are two problems: the bulging of the faces, whether it creates a shape like a saddle, and how you turn those bulging faces into multi-faceted shapes. The first is relatively easy to solve. The second is the main problem. Here one can draw hexagons on the side of the bulge, but these hexagons won’t be flat. The question is whether you can push and pull all these hexagons around to make each and everyone of them flat.”

During the imagined bulging process, even one that involves replacing the bulge with multiple hexagons, as Craven points out, there will be formation of internal angles. These angles formed between lines of the same faces – referred to as dihedral angle discrepancies – means that, according to Schein and Gayed, the shape is no longer a polyhedron. Instead they claimed to have found a way of making those angles zero, which makes all the faces flat, and what is left is a true convex polyhedron (see image below).

Their rules, they claim, can be applied to develop other classes of convex polyhedra. These shapes will be with more and more faces, and in that sense there should be an infinite variety of them.

Playing with shapes

Such mathematical discoveries don’t have immediate applications, but often many are found. For example, dome-shaped buildings are never circular in shape. Instead they are built like half-cut Goldberg polyhedra, consisting of many regular shapes that give more strength to the structure than using round-shaped construction material.

Only the one in the right bottom corner is a convex polyhedra. Stan Schein/PNAS

However, there may be some immediate applications. The new rules create polyhedra that have structures similar to viruses or fullerenes, a carbon isotope. The fact that there has been no “cure” against influenza, or common flu, shows that stopping viruses is hard. But if we are able to describe the structure of a virus accurately, we get a step closer to finding a way of fighting them.

If nothing else, Schein’s work will invoke mathematicians to find other interesting geometric shapes, now that regular convex polyhedra may have been done with.

The Conversation

This article was originally published on The Conversation.
Read the original article.


  1. Heres idea, what if light is in the form of the flower of life? Light was created when the flower was finished, wouldn’t that mean they are one and the same in some way? But light is just the lay out, life is the “electricity,” for this “ancient machine,” or at least, thats what i think, because the flower of life is no more then a map if it has nothing using it. Life is simply whatever fuels it, maybe its our term maybe its not, what was there before language can not be full interpreted with words, we can try though. Everything reflects everything, look at what we know and link it to what else we know then find what else we know of one side to find what may be true about the other. The flower of life can not do anything without conscious life, it is only a map on its own.

  2. Good article? Great article! Why isn’t this in all the newspapers all over the country?? Absolutely astounding. This represents one of the few great discoveries of the 21st century. I am in awe, absolutely speechless.

    1. Please do not judge something you don’t understand, become educated in the subject then have a friendly discussion with someone about it, not openly ridicule a group of people just because you can’t see past your own cultures.
      This is a website of learning not judgement

    2. It’s ok John. You don’t have to be so scared of stuff you don’t understand. If everyone is “so stupid”, what does that say about the guy who goes to a website that is “so stupid”, just to leave a comment?

  3. 400 years lol. I knew it without extensive research. But it’s cool nonetheless. You know I’d everyone would learn Reiki, there would be a lot of healing of viruses going on.

  4. To those if you thinking that he’s being sexist by saying “female” side if the brain, then you clearly haven’t seen many, or any of the spirit science videos. Specifically, spirit science 4 – male and female energy. If so then you would realize that there are no sexist meaning behind that statement. I emplore you all to check them out!

    1. There is no such thing as the ‘left brain, right brain’ thing. Its a myth, perpetuated by pseudo-scientific blogs like this one.

  5. Its the shape of the “seed” of life. More doors are opening towards the recovering of ancient knowledge. Keep it up!

  6. I think he meant the right side of the brain (female) Nothing sexist at all in that statement. You should look into that. Maybe you will have a better understanding. :)

  7. Can you not say “makes me wonder on a female type of way” ? Considering this is a “scientific” article, there are MUCH better ways of meaning what you say(that aren’t sexist)

    1. I think you mean that the scientific way is sexist. I don´t argue about that, but. Why should a woman argue with a man about using his right brain-side to discribe how females use to do themselves, on a scientific article?
      Sorry if you get confused.
      Best wishes.

  8. great article.
    The paper is unfortunately not open to… :)
    The mathematspresented this after a long time of research. in a mathematical way like leftside prayers. Now its about that the world picks it up see it in the world and to create with it. the right side has a different approach. one that seems unusual when trying to derive forumuals and set proofs together. well actually of course they are creative but within strict mathematical system.
    I would like to see a thounsand of generated shapes based on this new class.

    1. Hello, so pleased to see this.. because about 15 years ago i ATTENDED a seminar ,of DRUNVOLO MELCHIZEDEK; about the MERHKABA: AND THE FLOWER OF LIFE and we also studies the SHAPES you present here.. we were supposed to practice the triangles of Male and Female supperimposed on each other, so we trancsed upwards, but i could not do it…….until a Facilitator told us that some of us have to imagine the other SHAPES AROUND OUR BODY,,,,,,to my surprise…….i assended on the multi pointed ONE…What does this mean ?please let me know……. as that day everyone was saying something, and i lost my train of thought, LOVE AND LIGHT: Liliane