Exploring the Educational Value of Mathematical Beauty and Effective Ways of Discovering Mathematical Beauty

The precision of mathematical reasoning, the abstractness of mathematical language, the profundity of mathematical thought and method, as well as the excessive formalization of mathematics teaching have formed an impassable gap, hindering students in approaching mathematics. This has concealed the beauty of mathematics and the light of mathematical culture. However, if students are able to cross this gap, they would find that mathematics is a vast world full of vitality, imagination, wisdom, poetry, and beauty. The pursuit of mathematical beauty is one of the motivations for scientists to research this field. Experiencing mathematical beauty is of great significance to students’ learning and growth. In teaching, the value of mathematical beauty is explored, such as stimulating emotions, opening up to the truth, and cultivating goodness. Several effective ways are suggested in this article to guide students to discover the mathematical beauty in life while finding it in problem-solving methods and exploring it in knowledge systems.

The Platonic solids and fundamental tests of quantum mechanics

The Platonic solids is the name traditionally given to the five regular convex polyhedra, namely the tetrahedron, the octahedron, the cube, the icosahedron and the dodecahedron. Perhaps strongly boosted by the towering historical influence of their namesake, these beautiful solids have, in well over two millennia, transcended traditional boundaries and entered the stage in a range of disciplines. Examples include natural philosophy and mathematics from classical antiquity, scientific modeling during the days of the European scientific revolution and visual arts ranging from the renaissance to modernity. Motivated by mathematical beauty and a rich history, we consider the Platonic solids in the context of modern quantum mechanics. Specifically, we construct Bell inequalities whose maximal violations are achieved with measurements pointing to the vertices of the Platonic solids. These Platonic Bell inequalities are constructed only by inspecting the visible symmetries of the Platonic solids. We also construct Bell inequalities for more general polyhedra and find a Bell inequality that is more robust to noise than the celebrated Clauser-Horne-Shimony-Holt Bell inequality. Finally, we elaborate on the tension between mathematical beauty, which was our initial motivation, and experimental friendliness, which is necessary in all empirical sciences.

Look for beauty, because of Fibonacci numbers

“Look for beauty, because of Fibonacci numbers” explains how to use addition to construct an ordered list of numbers known as the Fibonacci numbers. The chapter explores several unexpected and lovely examples of these numbers in nature, including the numbers of clockwise and counterclockwise spirals in sunflowers, cacti, and pinecones. The discussion includes numerous photographs illustrating Fibonacci numbers in nature. Mathematics students and enthusiasts are encouraged to accept the daily invitation to practice searching for and finding abundant mathematical beauty around them. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Methodology to Prove the Quranic Correctness

I brainstormed about the methodology to prove the Quranic correctness i.e. to discover the components of Tahara I function, which advocates that studies by mathematical beauty, of the Quranic contents, and of the prophets are good to prove the Quranic correctness.

Super-Elastic Double-Helix Model of Photon. Huygens-de Broglie Particle on the Helical Path Guided by the Newton-Bohm Entangled Helical Evolute. Quantum of Magnetic Flux Based on the Mathematical Beauty of Newton, Lorentz, Einstein, Dirac, Gell-Mann, Schw

In our approach we have combined knowledge of Old Masters (working in this field before the year 1905), New Masters (working in this field after the year 1905) and Dissidents under the guidance of Louis de Broglie and David Bohm. In our model the photon is represented as the Huygens-de Broglie’s particle on the helical path (full wave) guided by the Newton-Bohm entangled evolute (empty wave). We have formulated the concept of the Super-Elastic Photon WAVE based on the Great Works of Weber, Abbe, Voigt and Einstein. This model works with the longitudinal elasticity of that WAVE that was already very well tested experimentally. Newly, we propose to test the elastic amplitude of this WAVE for the case of the Doppler’s redshift, the Doppler’s blueshift, and the Zwicky’s redshift. We have newly used the concept of the Lorentz’ force for the description of the photon acting force and the fermion reacting force. In this model the Lorentz’ factors γ and γ3 do not describe the “transverse mass of fermions” and longitudinal mass of fermions” but the “reacting transverse force of fermions” and the “reacting longitudinal force of fermions”. (The mass of photons and fermions does not change with their speed). It is very well-known that the cylindrical helix observed from different angles forms shadows in the Plato’s Cave as circle, sine, cosine, trochoid, cochleoid, hyperbolic spiral. Therefore, the resulting shape depends on the observer position in the Plato’s Cave-this is the famous Rashomon effect between observers. Based on the Newton-Bohm helical evolute and the Huygens-de Broglie helical path of the particle we have derived interesting formula known as the quantum of the magnetic flux. When we work further with this concept based on the Mathematical Beauty developed by Dirac, Gell-Mann, Schwinger, Polchinski, Witten and many others, we will obtain possible properties of the magnetic monopole. This photon quantum of the magnetic flux can be experimentally evaluated in the known tests with superconductors and micro-WAVES and infrared-WAVES. Can it be that Nature cleverly works with the magnetic monopole hidden in plain sight? We want to pass this concept into the hands of Readers of this Journal better educated in the Mathematics and Physics.