Logarithmic and Fibonacci Spirals in Plant Phyllotaxis

Nature, particularly in plants, features logarithmic and Fibonacci spirals, exemplifying the elegance of natural design and the rhythmic dance of life, encompassing symmetry and other intriguing mathematical phenomena, including recursive functions.

Spiral patterns in plants emerge from their repetitive growth, where each turn closely mirrors the previous one with scaling or rotational adjustments. This growth process, common in nature and known as phyllotaxis, utilizes recursive functions, which can generate logarithmic and Fibonacci spiral patterns.

The Fascinating World of Runic Calendars

The Runic calendar, also referred to as a Rune almanac, served as a perpetual timekeeping tool throughout Northern Europe until the 19th century. Structured with lines of symbols, it marked significant astronomical events and celebrations, including solstices, equinoxes, and Christian holidays. These symbols were often etched onto parchment or carved into various materials such as wood, bone, or horn.

One of the most esteemed examples of these calendars is Worm’s Norwegian runic calendar from 1643, renowned for its bone craftsmanship. Danish Antiquarian Ole Worm featured it in his book “Fasti Danici, universam tempora computandi rationem antiquitus in Dania et vicinis regionibus observatam libris tribus exhibentes.” Although he extensively detailed the winter months in his work, he omitted details regarding the summer season. Fortunately, supplementary insights are provided through ‘runstavs’ and ‘primstavs.’ ‘Runstavs’ served as runic sticks used in divination practices, while ‘primstavs’ were Norwegian wooden calendar sticks primarily employed for timekeeping and weather prediction.

runic calendar


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Beyond 65 digits, π serves no practical purpose

For spatial engineers’ highest accuracy calculations, used in interplanetary navigation, 3.141592653589793 is more than sufficient. Let’s understand why more decimals aren’t needed.

Consider these examples:

• Voyager 1, the farthest spacecraft from Earth, is about 14.7 billion miles away. Using π rounded to the 15th decimal, the circumference of a circle with a radius of 30 billion miles would be off by less than half an inch.

• Earth’s circumference is roughly 24,900 miles. The discrepancy using limited π would be smaller than the size of a molecule, over 30,000 times thinner than a hair.

• The radius of the universe is about 46 billion light years. To calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom, only 37 decimal places are necessary.

• With just 65 decimal places, we could determine the size of the observable universe within a Planck length, the shortest measurable distance.

While π’s digits are endless, for microscopic, macroscopic or cosmic endeavors, very few are necessary.

Illusion vs Reality

“Illusion, a derivative of reality, and vice versa.” – GS

For a little backstory… one day, a follower threw me a curveball: ‘What separates illusion from reality?’ I countered with a snap response: ‘What separates acceleration from speed?’

Wandering Eye

The entire sea urchin functions as a massive compound eye because each of its spines conceals tube feet with light-sensitive cells at their bases. Essentially, a sea urchin is one large, moving, spine-covered eye. While its vision might not astonish an eye doctor, for an animal devoid of actual eyes, it’s rather impressive!

A representation of the computational model of the ‘spherical’ vision of the sea urchin.

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Visual Maths


Mathematical objects like spirals and fractals can serve as art. Their deceptively simple nature adds an extra allure to this artistic expression.

Explore the realm of Visual Mathematics, a distinctive fusion of Art and Math. Delve into intricate diagrams, tables, and visually captivating pieces that provide a delightful experience for both the eye and the mind. Visit our online gallery for prints, posters, and t-shirts showcasing beautiful math art.

Math and Art
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The Golden Ratio is implemented in circles, serving as a mathematical ratio for generating aesthetically pleasing designs. Given its prevalence in nature, the natural appearance of outcomes is unsurprising.

Golden ratio with circles
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The Golden Diamond presents an asymptotic monomorphic dissection of the equilateral triangle. Each tile follows proportions aligned with the golden ratio in relation to the outer triangle.

Golden ratio with equilateral triangles
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Golden Spiral created using equilateral triangles. The Padovan spiral follows a recursive sequence akin to the Fibonacci sequence.

Padovan spiral
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The Geometry of the Bees

When constructing a honeycomb, bees aim to minimize wax usage and honey consumption, using the least wax necessary for a comb with maximum honey storage. The wax cells are designed with interlocking opposing layers, sharing facets at closed ends while having open ends facing outwards (see fig. 1). Each cell is a ten-sided structure with a rhombic decahedron form – a hexagonal prism with three rhombi at its closed end (fig. 2). Mathematicians have extensively studied the highly efficient isoperimetric properties of these cells. The question remains: What angle alpha maximizes volume while minimizing surface area on each cell face when the hexagonal prism’s faces have a width of 1 unit?

Continue reading “The Geometry of the Bees”

Fibonacci Zoetropes

The Fibonacci Zoetropes are sculptures by John Edmark. The spirals in the sculptures follow the Fibonacci sequence. When filmed at 24 frames per second and spun at 550 revolutions per minute, each frame represents a 137.5 degree rotation, which is equivalent to the Golden Angle.

© John Edmark