Let’s dive into the fascinating world of Johannes Kepler, a key figure in science history, especially in astronomy and geometry. His groundbreaking work helped us understand complex patterns in nature, paving the way for the discovery of aperiodic tilings. These tilings challenge our traditional ideas about symmetry and structure.
Johannes Kepler is famous for discovering that planets orbit the sun in elliptical paths, not circular ones. Before this, he imagined the solar system as a series of nested spheres separated by the five Platonic solids: the cube, tetrahedron, octahedron, dodecahedron, and icosahedron. This model was his attempt to find geometric harmony in the universe, a concept he strongly believed in.
Kepler’s love for geometry also led him to practical problems, like how to stack cannonballs efficiently. In 1611, he introduced what we now call Kepler’s conjecture, suggesting that the best way to stack spheres is through hexagonal close packing. Although he stated this as a fact, it took nearly 400 years for mathematicians to prove it, with the proof finally published in 2017.
In his pamphlet “De Niva Sexangula” (On the Six-Cornered Snowflake), Kepler wondered why snowflakes always have a six-cornered star shape. Even before atomic theory was developed, he speculated about tiny units, similar to molecules, that might form such structures. This was an early hint at the science of crystallography.
Kepler’s exploration of geometric patterns led him to study how shapes can tile a flat surface. He found that regular hexagons could tile a plane perfectly, but regular pentagons could not. Despite his efforts, he couldn’t find a way to tile the plane periodically using pentagons.
In 1961, mathematician Hao Wang studied multi-colored square tiles and guessed that if a set of tiles could tile the plane, it would do so periodically. However, his student Robert Berger found a set of 20,426 tiles that could only tile the plane non-periodically, introducing the idea of aperiodic tiling.
Mathematicians aimed to find smaller sets of tiles for aperiodic tiling. Berger reduced the number to 104 tiles, and Donald Knuth further reduced it to 92. In 1969, Raphael Robinson discovered a set of just six tiles, but it was Roger Penrose who simplified it to two tiles—a thick rhombus and a thin rhombus.
Penrose’s work with these two tiles allowed for the creation of aperiodic patterns with five-fold symmetry. He showed that these patterns could extend infinitely without repeating, offering a new perspective on symmetry in mathematics.
The impact of Penrose tilings went beyond mathematics into materials science. In the early 1980s, Paul Steinhardt and his students modeled how atoms could arrange themselves in structures like Penrose tilings, leading to the discovery of quasicrystals. These materials defied traditional crystallography, showing long-range order without periodicity.
Initially, the scientific community was skeptical, with Linus Pauling dismissing quasicrystals as “quasi-science.” However, the discovery was validated when Dan Schechtman received the Nobel Prize in Chemistry in 2011 for his work on quasicrystals. These materials are now used in various fields, including non-stick coatings and durable steel.
From Kepler’s geometric models to the discovery of aperiodic tilings and quasicrystals, this journey shows the evolution of scientific thought. It highlights how ideas once thought impossible can lead to groundbreaking discoveries, reshaping our understanding of the natural world. The interplay between mathematics and materials science continues to inspire exploration into the hidden patterns that govern our universe.
Using a string and two pins, create an ellipse on paper to understand Kepler’s first law of planetary motion. Measure the distances from the foci to any point on the ellipse and verify that their sum is constant. Discuss how this relates to the orbits of planets around the sun.
Simulate the packing of spheres using small balls (like marbles) and a container. Try different packing methods to see which allows the most efficient use of space. Discuss how Kepler’s conjecture applies to modern packing problems in logistics and storage.
Examine real or paper snowflakes to identify their six-fold symmetry. Create your own snowflake designs using paper folding and cutting techniques. Discuss how Kepler’s early thoughts on snowflakes relate to modern crystallography and molecular structures.
Using cut-out shapes of Penrose tiles (thick and thin rhombuses), attempt to cover a flat surface without creating a repeating pattern. Explore the concept of aperiodic tiling and discuss its implications in mathematics and materials science.
Build a model of a quasicrystal using molecular model kits or 3D printed parts. Discuss how these structures differ from traditional crystals and explore their applications in modern technology, such as non-stick coatings and durable materials.
Geometry – The branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids. – In our geometry class, we learned how to calculate the area of a circle using the formula $A = pi r^2$.
Patterns – Repeated or recurring sequences or designs, often used to identify mathematical relationships or predict future outcomes. – By observing the patterns in the sequence of numbers, we were able to deduce the formula for the $n$th term.
Symmetry – A property where a shape or object is invariant under certain transformations, such as reflection, rotation, or translation. – The butterfly’s wings exhibit bilateral symmetry, meaning one half is a mirror image of the other.
Tiling – The covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. – The artist used a tiling pattern of hexagons to create a visually appealing mosaic on the floor.
Conjecture – An unproven statement or hypothesis that is based on observations and appears to be true. – The mathematician proposed a conjecture about the distribution of prime numbers that has yet to be proven.
Aperiodic – Describing a pattern that does not repeat at regular intervals. – The Penrose tiling is an example of an aperiodic tiling, as it never repeats itself exactly.
Hexagonal – Relating to a six-sided polygon or having a shape that resembles a hexagon. – The honeycomb structure is hexagonal, providing strength and efficiency in the use of space.
Crystalline – Having the structure and form of a crystal; composed of a repeating pattern of atoms or molecules. – The crystalline structure of salt can be observed under a microscope, revealing its cubic symmetry.
Orbits – The paths that objects follow as they move around a central point or body, often described by mathematical equations. – The planets in our solar system have elliptical orbits, as described by Kepler’s laws of planetary motion.
Shapes – Forms or outlines of objects, which can be described mathematically by their properties such as angles, sides, and symmetry. – In geometry, we study various shapes, including polygons, circles, and ellipses, to understand their properties and relationships.
