#geometry matters

The Unexpected Fractal Signatures in Fibonacci Chains

Fang Fang *, Raymond Aschheim and Klee Irwin

Abstract

In this paper, a new fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L/S = f.

The corresponding pointwise dimension is 1.7. Various modifications, such as truncation from the head or tail, scrambling the orders of the sequence and changing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to changes in the Fibonacci order but not to the L/S ratio.

Introduction Quasicrystals possess exotic and sometimes anomalous properties that have interested the scientific community since their discovery by Shechtman in 198. Of particular interest in this manuscript is the self similar property of quasicrystals that links them to fractal. Historically, research on the fractal aspect of quasicrystalline properties has revolved around spectral and wave function analysis. Mathematical investigatio of the geometric structure of quasicrystals is less represented in the literature than experimental work.

In this paper, a new framework for analyzing the fractal nature of quasicrystals is introduced.

Specifically, the fractal properties of a one-dimensional Fibonacci chain and its variations are studied in the complex Fourier space. The results may also be found in two and three dimensional

Study shows we like our math like we like our art: Beautiful

by Kendall Teare, Yale University

A beautiful landscape painting, a beautiful piano sonata—art and music are almost exclusively described in terms of aesthetics, but what about math? Beyond useful or brilliant, can an abstract idea be considered beautiful?

Yes, actually—and not just by mathematicians, reports a new study in Cognition.

Coauthored by a Yale mathematician and a University of Bath psychologist, the study shows that average Americans can assess mathematical arguments for beauty just as they can pieces of art or music. The beauty they discerned about the math was not one-dimensional either: Using nine criteria for beauty—such as elegance, intricacy, universality, etc.—300 individuals had better-than-chance agreement about the specific ways that four different proofs were beautiful.

This inquiry into the aesthetics of mathematics began when study co-author and Yale assistant professor of mathematics Stefan Steinerberger likened a proof he was teaching to a "really good Schubert sonata."

"As it turns out, the Yale students who do math also do a statistically impressive amount of music," said Steinerberger. "Three or four students came up to me afterwards and asked, 'What did you mean by this?' And I realized I had no idea what I meant, but it just sounded sort of right.”

The fourth dimension

by Hinton, Charles Howard, 1853-1907

A four-dimensional figure, pag 135

These separate figures are the successive stages in which the whole four-dimensional figure in which they cohere can be apprehended. The first figure and the last are tetrakaidecagons. These are two of the solid boundaries of the figure. The other solid boundaries can be traced easily. This kind of boundary is hexagonal prism. Other solid boundaries can be traced through four of the five sectional figures. These four sections are the sections of a tetrakaidecagon as can be recognised from the sections of this figure we have had previously. Hence the boundaries are of two kinds, hexagonal prisms and tetrakaidecagons.

Fractal solar panels: Optimizing aesthetic and electrical performances

Ellis T. Roe, Alexander J. BiesID, Rick D. Montgomery, William J. Watterson, Blake Parris, Cooper R. Boydston, Margaret E. Sereno, Richard P. Taylor

Abstract Solar energy technologies have been plagued by their limited visual appeal. Because the electrical power generated by solar panels increases with their surface area and therefore their occupancy of the observer’s visual field, aesthetics will play an increasingly critical role in their future success in urban environments. Inspired by previous psychology research highlighting the aesthetic qualities of fractal patterns, we investigated panel designs featuring fractal electrodes. We conducted behavioral studies which compared observers’ preferences for fractal and conventional bus-bar electrode patterns, along with computer simulations which compared their electrical performances. This led us to develop a hybrid electrode pattern which best combines the fractal and bus-bar designs. Here we show that the new hybrid electrode matches the electrical performance of bus-bars in terms of light transmission and minimizing electrical power losses, while benefiting from the superior aesthetics of fractal patterns. This innovative integration of psychology and engineering studies provides a framework for developing novel electrode patterns with increased implementation and acceptance.

Introduction The cost of solar panels has more than halved since 2010, triggering a rise in their popularity. For example, the California Energy Commission recently voted to make rooftop solar panels a mandatory addition to all new single-family homes commencing in 2020. However, their large surface areas will significantly impact our daily visual experiences. The psychology of aesthetics will therefore play an increasingly critical role in solar panels’ success, especially as aesthetics has been shown to be a major concern when considering solar technologies. Recent ‘blended’ designs seek to neutralize the poor aesthetics of traditional panels by covering them with camouflaging louvers which visually match the panels to their surroundings. Rather than neutralizing their visual impact, we propose panels that actively enhance the environmental aesthetics.

Solar technology has come a long way since the first practical silicon solar cell was developed at Bell Labs in 1954. Today’s panels typically feature an array of 10x10cm solar cells which employ electrodes on their top surface to transmit electricity generated by the underlying photodiode to an external circuit [6–7]. The traditional ‘bus-bar’ electrode design features horizontal ‘fingers’ (each 50–200μm wide) intersected by several ‘bars’ (1-2mm wide). The number of fingers and bars vary between designs with the goal of optimizing their optical and electrical efficiencies. Some designs

The Haunting Beauty of Snowflakes: Wilson Bentley’s Pioneering 19th-Century Photomicroscopy of Snow Crystals

by Maria Popova on Brain Pickings

The quest to capture nature’s vanishing masterpieces, endowed with the delicacy of flowers and the mathematical precision of honeycombs.

Wilson Bentley (February 9, 1865–December 23, 1931) was fifteen when his mother, aware of her son’s sensitive curiosity and artistic bent, strained the family’s means to give him a microscope for his birthday. Over the next four years, while Walt Whitman was exulting a state over that “a leaf of grass is no less than the journey work of the stars,” Wilson placed every curio he could find under his microscope: blades of grass, pebbles, insects. The day he managed to place a snowflake on the glass plate and to savor its microscopic perfection before it melted, he was besotted. Snowflakes became his life. “Miracles of beauty,” he called them. He began sketching what he saw through his microscope, but felt that his drawings failed to capture the full miraculousness before it vanished into liquid erasure. Although his father was already irate with the boy’s artistic deviation from farm labor, “fussing with snowflakes” rather than pulling potatoes, Wilson somehow persuaded him to invest in a camera.

Weeks before his twentieth birthday, he mounted his new 1.5-inch microscope eyepiece to the lens of his enormous view camera with its accordion-like body fully extended. On January 15, 1880, Wilson Bentley took his first photograph of a snowflake. Mesmerized by the beauty of the result, he transported his equipment to the unheated wooden shed behind the farmhouse and began recording his work in two separate sets of notebooks — one filled with sketches and dedicated to refining his artistic photomicroscopy; the other filled with weather data, carefully monitoring the conditions under which various snowflakes were captured.

The chords of the Universe

by Eli Maor for Aeon

It’s no surprise that mathematics has influenced music. But did you know that the influence goes both ways?

Music and mathematics have always been intimately intertwined. Anyone who has ever played a musical instrument is aware of the presence of mathematics on every page of the score – from the time signature that sets a piece’s rhythm, to the metronome number that determines the speed at which the piece should be played; and, of course, the very act of playing music requires us to count 1, 2, 3… and arrange these numbers into groups, called bars or measures. So it comes as no surprise that mathematics has had a significant influence on music. Much less known is that the influence extends both ways.

The Greek philosopher Pythagoras, active during the 6th century BCE, might have been the first to uncover a quantitative relation between music and mathematics. Experimenting with taut strings, he found that shortening the effective length of a string to one-half its original length raises the pitch of its sound by an agreeable interval, an octave. Other ratios of string lengths produce smaller intervals: 2:3 corresponds to the musical interval of a fifth (so called because it is the fifth note up the scale from the base note); 3:4 corresponded to a fourth; and so on. Pythagoras also discovered that multiplying two ratios is equivalent to adding their intervals: (2:3) x (3:4) = 1:2, so a fifth plus a fourth equals an octave. In doing so, he unknowingly came up with the first logarithmic law in history.

The octave, the fifth and the fourth produced pleasing combinations of tones, or consonants, whereas more complicated ratios, such as 9:8 or 15:16, led to dissonances. This revelation made a deep impression on the Pythagoreans, prompting their belief that everything in the Universe – from the laws of musical harmony to the motion of the Sun, Moon and the five planets – was governed by simple ratios of whole numbers. Number rules the Universe was the Pythagorean motto. It would dominate scientific thought for the next two millennia.

When Magic Is Seen in Twisted Graphene, That’s a Moiré

by  Pradeep Mutalik for Quanta Magazine

When two very similar grids with light and dark elements overlap, new sinuous patterns emerge that seem to shimmer and flow. Whether you’re into art or science, engineering or fashion, you have probably seen or at least heard of these moiré patterns. The name moiré (pronounced mwa-ray) is etymologically related to the French word for mohair and entered the language centuries ago, describing rippled or watered fabrics. Today, moiré patterns are seen not only in visual fields such as optics, art, photography and color printing, but also in seemingly far-flung areas like marine engineering and the detection of forged banknotes.

Recently, moiré effects have also entered the quantum realm in a big way, as David H. Freedman reported in his Quanta article “With a Simple Twist, a ‘Magic’ Material Is Now the Big Thing in Physics.” Freedman highlights a stunning discovery made by the experimental physicist Pablo Jarillo-Herrero and his colleagues at the Massachusetts Institute of Technology: When a layer of graphene, a sheet of carbon crystal with atoms arranged in a hexagonal one-atom-thick lattice, is dropped on another one and rotated to just the correct angle of about 1.1 degrees, the graphene magically acquires the ability to become superconductive when the requisite number of electrons are added. That interesting things would happen at around this magic angle had been mathematically predicted by the theorists Rafi Bistritzer and Allan H. MacDonald, both then at the University of Texas, Austin. In fact, Jarillo-Herrero’s team made their discovery while trying to create graphene twists that matched a bunch of magic rotation angles from this earlier prediction, including 1.05 degrees, 0.5 degrees and 0.24 degrees.

Water Seems to Have Not One, But Two Distinct Structures in Its Liquid State

In 2018, scientists from the UK and Japan demonstrated that these peculiar properties have to do with the tetrahedral arrangement of water molecules in liquid form. This means that every water molecule is hydrogen-bonded to four others in a rough pyramid shape.

But still, how the structure is ordered has remained a topic of debate. One model proposes that water's molecular structure is unimodal - it's tetrahedrons all the way down. The other proposes that the structure is bimodal, consisting of two structures - tetrahedrons and something else.

To try and resolve the issue, industrial scientists from the University of Tokyo conducted computer simulations, and also ran experiments on liquid silica, one of the few other liquids known to have a tetrahedral molecular arrangement as well.

"We show the first clear numerical evidence in the structure factor for the dynamical coexistence of the two types of local structures ... supporting the two-state description of liquid water," the researchers wrote in their paper.

continue on sciencealert.com

Artilcle abstract:

Water is the essential liquid on earth since it not only plays vital roles in living systems but also has a significant impact on our daily life from various industrial applications to earth’s climate system. However, the unusual properties of liquid water, if compared with other liquids, has puzzled us for centuries because the basic structure of liquid water has remained unclear and has continued to be a matter of serious debate. Here, by computer simulations of three popular water models and the analysis of recent scattering experimental data, we show that there are two overlapped peaks hidden in the apparent “first diffraction peak” of the structure factor. One of them (ordinary peak) corresponds to the neighboring O–O distance as in ordinary liquids, and the other (anomalous peak) corresponds to a longer distance. We reveal that this anomalous peak arises from the most extended period of density wave associated with a tetrahedral water structure and is to be identified as the so-called first sharp diffraction peak that is commonly observed in silica and other tetrahedral liquids. In contrast, the ordinary peak arises from the density wave characteristic of local structures lacking tetrahedral symmetry. This finding unambiguously proves the coexistence of two types of local structures in liquid water. Our findings not only provide vital clues to settle a long-standing controversy on the water structure but also allow direct experimental access to the fraction of tetrahedral structures in liquid water.