‘Geometry and Joyce at Pennsylvania lodge’

    

Be sure to attend The Pennsylvania Lodge of Research’s meeting in June at Williamsport.

From the summons:

You are hereby summoned to a stated meeting of the Pennsylvania Lodge of Research to be held on Saturday, June 15, 2024, at Williamsport Masonic Lodge, 360 Market St., Williamsport, PA 17701, beginning at 10:00 o’clock ante meridian, Eastern Time. A luncheon will be held following the meeting, at approximately 12:00 p.m. (Reserve here.)

Presentations:

Bro. Theodore Schick, PM, Fellow of the Lodge of Research: “How Geometry Demonstrates the More Important Truths of Morality.”

Bro. J.L. Pearl: “Craftygild Pageantries: a Masonic Introduction to James Joyce’s Finnegans Wake.”

There was talk at New Jersey’s research lodge about carpooling to this meeting. Usually our Saturdays coincide, making visitation impossible, but not this time.

     

‘Free class: Mathematics and Logic’

    

Hillsdale College

“We do a disservice to young people by not teaching them all the steps in mathematics, because it’s liberating to know. And by discarding Euclid, you have cut them off from a way to understand logic, from which all accurate reasoning comes. And you have cut them off from also understanding the beauty of the relationships in nature.”

Larry P. Arnn, President

Hillsdale College

Hillsdale College offers another free online course to the world: “Mathematics and Logic: From Euclid to Modern Geometry.” From the publicity:

Today, more than ever, we need logic and sound reasoning in defense of truth. And one of the best ways to develop the skills is through the study of Euclidean Geometry.

Hillsdale College

For more than 2,300 years, Euclid’s Elements has provided the foundation for countless students to learn how to reason with precision and pursue knowledge in all fields of learning. This classic text of Western civilization provides profound tools to distinguish truth from error by means of self-evident principles.

In this course, you will study the transformation of mathematics by the ancient Greeks, discover the fundamentals of logic and deductive reasoning, examine the central proofs of Euclid, learn about the birth of modern geometry, and much more.

Hillsdale College

By enrolling in this free online course, you’ll receive access to eleven lectures by Hillsdale’s distinguished mathematics faculty, a course study guide, readings, a course discussion board, and quizzes to aid you in the examination of the fundamentals of good mathematics.

Click here to get going. 

     

‘Geometry Salons at Anthroposophy NYC’

     
Saturday afternoon, the Anthroposophical Society of New York City will host a talk on Vedic Squares. This will be the first in a series running through next August. From the publicity:

Geometry Salon:

Vedic Squares

Saturday, September 14

at 1:30 p.m.

Anthroposophical Society of NYC

138 West 15th Street, Manhattan

Suggested donation: $5

The Vedic Square is a variation on a typical 9×9 multiplication table, which is a source of many Islamic patterns and symmetric art patterns.

Steve Pomerantz

The Geometry Salon meets monthly to explore the intersection of Art and Geometry. Topics have included Islamic and Cosmatesque Design, Projective Geometry, Classical Constructions, Form Drawing, and more. Our main presenters are Steve Pomerantz, John Lloyd, and Steve Bass. We rotate facilitators to lead discussions and drawing through a range of examples taken from history.

Steve Pomerantz

Accessible to people with all levels of drawing experience. Bring your imagination—and a ruler, compass, paper, pencils. Additional materials that could be useful include: colored pencils, watercolors, and some paper at least 8½ x 11. We will have some extra supplies for people who need them.

We will meet in a beautiful sunny room generously provided by Anthroposophy NYC. We are requesting a $5 contribution from everyone to cover the use of the space.

Future dates: October 12, November 9, December 21, January 11, February 22, March 14, April 18, May 16, June 6, July 11, and August 8.

Steve Pomerantz

At September’s meeting, we will be drawing patterns based on magic squares from the Vedic Tradition, which John Lloyd learned from Pieter Weltevrede and Mavis Gewant.

In October, Steve Pomerantz will show us how to draw Cosmati patterns. In November, Steve Bass will be showing us how to draw Rose Windows. In December, Steve Pomerantz will continue to show us how to develop Cosmati patterns. In January, Kelly Beekman will be show how to draw the planetary seals of Rudolf Steiner (based on a seven pointed star). In the Spring of 2020, we will explore patterns from Islamic Cultures and learn how to draw the Shri Yantra from the Vedic Tradition.

Geometry images below are by Steve Pomerantz.

‘The Hidden Business of Masonic Building’

     

No, this isn’t about a dodgy trustees meeting. Earnest Hudson, Jr., Worshipful Master of Joseph Warren-Gothic Lodge No. 934 in the Seventh Manhattan District, will visit New Jersey next week for a speaking engagement.

 Magpie file photo

A Matter of Geometry:

The Hidden Business

of Masonic Building

Peninsula Masonic Lodge No. 99

888 Avenue C

Bayonne, New Jersey

Lodge to open at 6:30

Dinner & Lecture at 7:30

Open to Masters and Fellows

RSVP here

I’m really looking forward to this.
     

‘Goldberg variations: the new class of polyhedra’

     
If you are the kind of thinker who hears the Divine in the language of geometry, then this news is for you. The National Academy of Sciences of the United States of America has just published a paper it received for review last spring that gives the world a new—fourth—class of convex polyhedra. In short, Plato, Archimedes, and Kepler have company, and his name is Goldberg. As you’ll see in the article below, these Goldberg variations (sorry, I couldn’t resist) are not true polyhedra solids, so “Goldberg” is a misnomer, albeit a well-intentioned one. It is believed this discovery can bring researchers closer to finding cures for a variety of viruses, if you’re curious about practical significance.

It’s not every day that something like this pops up, so for only the second time in Magpie history I’m going to reproduce an entire news story—replete with art—here, with thanks to The Conversation, the “academic rigor, journalistic flair” journal you all should have bookmarked for reference. Enjoy, and please discuss among yourselves.

After 400 years, mathematicians find

a new class of solid shapes

Not so special anymore.

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 love for geometry

Equilateral 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.

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.

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.

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.

However, there may be some immediate applications. The new rules create polyhedra that have structures similar to viruses or fullerenes, a carbon allotrope. 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 equilateral convex polyhedra may have been done with.