Throughout history, thinkers from various disciplines have given Platonic polyhedrons (regular polyhedrons) almost mysterious properties. Think that they hide the mysteries of the depths of the universe. They seem very simple, but they always give people an invisible feeling; humans have spent thousands of years exploring polyhedrons. They, especially the three-dimensional Platonic polyhedrons, make us feel happy and at the same time bring us endless confusion. They also have a natural attraction to artists, who often sit on these geometric cornerstones and meditate, looking for inspiration and mystery.

The art of mathematics for the masses

Mathematical sculptor George Hart said that mathematics is a science of observing patterns. Patterns can be seen everywhere: in music, in photos, in buildings, the shape of a crystal or the tilt of the earth’s orbit is also a pattern. Adults often have to learn to use mathematics to look at the world rationally, but Hart said that children have an advantage, that is, they look at the world naturally and honestly. But as they grow older, they will lose this advantage.

Hart often conducts sculptural and architectural exploration activities, leading his audiences, even kindergarten children, into various corners of the mathematical landscape, such as topology and solid geometry. Hart used his sculptures and hand workshops to challenge the teaching methods of mathematics. Many people associate mathematics with painful training in multiplication and rote geometric proofs. Students usually do not feel the beauty, elegance and magic of this subject. Until entering the university mathematics study. By then, it is often too late. Many people who might become mathematicians have already chosen other fields.

Harbin visited high schools and universities around the world during a certain period of time. When we interviewed, he had just returned from five schools in Vietnam, where he gave lectures and helped the team there build customized large-scale geometric sculptures. For a long time after his departure, these sculptures will stand there. These projects combine team building with self-study math classes.

Throughout Hart’s life, he has been building geometric sculptures using materials he found around. He remembers that when he was a child, his first work was a symmetrical strange thing made of many teeth. He has used knives, forks, spoons, compact discs, floppy disks, pencils, paper clips, brushes and toothbrushes to make three-dimensional artworks. Most of his works are derived from Platonic polyhedrons, which are shapes composed of polygons on a two-dimensional plane connected by edges and vertices.

Platonic Polyhedron

He said: “In a sense, I always start with mathematics that interest me and try to turn it into a physical thing.” In the early 1990s, he began to create sculptures when he was still a long island stone. A professor of communication theory and engineering at Brook University. Sculpture was just a hobby at the beginning. He said, “Later, it gradually became an obsession.” In 1999, the Walpar Gallery in New York City was exhibiting MC Escher’s work, and the exhibitor agreed to exhibit at the same time. Some of Hart’s works. He said: “It makes me realize that I am a sculptor.”

George Hart’s sculpture “Champ”, 4 feet in diameter (approximately 1.2 meters), is placed at Boamburton College in Manchester, Vermont. Its 30 identical components mimic a group of sea monsters-the legendary creatures of Lake Champlain.

Now he has a home studio filled with tools such as grinding wheels and saws, as well as laser cutters and 3D printers. His tool library also includes some tools he made himself, which can fix, bend and change materials according to his requirements, without any ready-made equipment.

Hart is not only keen to cultivate the passion for mathematics of the next generation, he also hopes that they will discover mathematics by themselves. Due to the cost of materials and other factors, the cost of his sculpture production may be as high as tens of thousands of dollars, and public schools often have no budget. In the past few years, he and mathematics educator Elisabeth Heathfield (Elisabeth Heathfield) have been working on projects to make sculpture cheaper, so that teachers can afford it within the classroom budget, and even use downloadable videos. The project was named “Mathematically Visualize”.

He is also a pioneer in another project that strives to expand the appeal of mathematics. Hart helped create the “Museum of Mathematics”, the only mathematics museum in the United States, which opened in New York City in 2012. Hart spent five years on its design and content. Since its opening, more than 500,000 people have visited this museum.

Hart hopes that the “Museum of Mathematics” is just the beginning of the “normalization” trend of mathematics. He hopes that mathematics is as popular as our language. He and Heathfield envisioned a worldwide network of mathematics centers, which functions similar to a library. As a public space, people can gather online to learn and share ideas about mathematics.

The Mathematics Behind Art: The Polyhedron Family

Platonic polyhedrons are very special. Only five regular polyhedrons can exist in three-dimensional space. They include a cube, which has 6 square faces; a regular tetrahedron, which has 4 regular triangular faces; also includes a regular octahedron, which has 8 regular triangular faces; a regular icosahedron, which has 20 regular triangular faces; finally It is a regular dodecahedron with 12 faces, all of which are regular pentagons.

For thousands of years, people have been praising, analyzing, transforming, and even sacred these forms. They are really special because they are the only five types of regular convex polyhedrons. First, their faces must be regular polygons, and they must be mutually congruent. (This means that they are the same size and shape.) Second, each vertex has the same number of faces intersecting here.

Platonic solids we call them, because the Greek astronomer Plato around 360 BC in which the

work “Timaeus” (Timaeus) described them, but the Greeks are not the only people who know them. Some scholars believe that some carved stones have been found in Scotland to take the shape of a Platonic polyhedron, each of which is the size of a baseball. This indicates that some ancient cultures may have known the Platonic polyhedron at least a thousand years before Plato.

Plato not only described these polyhedra, he also admired them very much. He connected four of them with the four basic elements “fire, earth, water, and air” that constituted the world at that time. The regular tetrahedron represents “fire”; the cube represents “earth”; the “water” is represented by the regular icosahedron; the “qi” corresponds to the regular octahedron. Regarding the remaining regular dodecahedrons, Plato wrote in the “Timeou”: “God uses regular dodecahedrons to arrange the order of the entire sky constellation.”

In “Geometric Elements”, Euclid proved it. There can only be five kinds of Platonic polyhedra, there can be no more. Plato associates the regular polyhedron with sacred geometry, but he is not the only one to do so. In 1596, the German mathematician and physicist Johannes Kepler published the book “Mysteries of the Universe” when he was only 24 years old. He believed that God built the solar system based on Platonic polyhedrons.

At that time, astronomers knew only six planets. Kepler wanted to know why there were six planets and why they would orbit the sun in a staggered way along increasing orbits. Plato’s polyhedron gave him an answer. Kepler relied purely on his imagination to associate the planetary orbit with the regular polyhedron without any evidence.

Imagine you draw a circumscribed sphere around each Platonic polyhedron. Kepler pointed out that by nesting these spheres and regular polyhedrons together in a specific (but seemingly random) order, you can get a geometric configuration in which the Platonic polyhedrons The ratio of distance to the distance between planetary orbits is exactly the same. Kepler thought it was a fascinating bowl. “Earth’s orbit is the measure of all things; it’s track box to live in a sphere of its orbit sphere is a dodecahedron, and surround the sphere surface dodecahedron is Mars

orbit; then To the outside, the sphere that frames the orbit of Mars is a regular tetrahedron,” Kepler wrote. “Now you know the number of planets

mesh is the reason of the six.” I believe Platonic solids and Kepler orbits of the planets, as well as all proportion in nature, are determined by God.

Kepler’s research is based on religious pursuits in many ways. But in 2011, astronomer Kenneth Brecher of Boston University pointed out that viewing “Mysteries of the Universe” as an important turning point in scientific research was for other reasons. Kepler wanted to know the significance of this arrangement of planetary orbits. He went beyond observation and boldly used mathematical causality. Although he adjusted his Plato polyhedron model until it roughly aligned with the planet’s orbit, this was a coincidence at best.

13 types of Archimedes polyhedra

For centuries, mathematicians have used other methods to expand and explore Platonic polyhedra. For example, you can make a star expansion. You can select a face of a regular polyhedron, and then extend the plane of its adjacent face until they intersect. Do the same for all faces.

There are other polyhedron families. There are 13 types of Archimedes polyhedrons, and their sides all have the same length, but are composed of two or more regular polygons. Like Platonic polyhedra, they have a high degree of symmetry, which makes them attractive as people who study group theory and crystal shapes.

For a long time, artists have been fascinated by these shapes, which are the themes of both mathematics and art. Even Da Vinci loved to scribble polyhedrons in his notebooks. (Da Vinci is very interested in geometry. He recommended a style of depicting polyhedrons, which is to draw only their edges, so that the observer can see through the polyhedrons and see how their vertices are connected.) The charm of Platonic polyhedrons It lasted for centuries, and for artists like Hart, they found a new way to interpret the classics, showing the eternity of Plato’s polyhedrons.