#euler spiral

[art-of-mathematics]

Icosahedron net - "coiling up"

This icosahedron net is really cool, as it is just like a "string of triangles" you can just roll up to form the icosahedron.

Result:

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The hexagon grid is really helpful for drawing equilateral triangles.

- - -- ---

It reminds me of this:

Diffraction - Source: German PDF [found at TU-dresden website]

...

[arbieroo-art]

Euler spirals.

[inksandpensblog]

[me, seeing the AvGeometry thumbnail]: they’re finally doing Fibonacci

(also I must say that desk mats are the most appropriate merch for this series ever)

okay there’s no excuse not to make fanart anymore now that we know we can just make a single colored line, we don’t even have to make a whole stickfigure this is taking “golden ratio” literally

oh neat so that’s why φ moved like that in AvMath. I’d thought it seemed like it was moving on a line and I was right, we just couldn’t see the line then

this is when I realized how happy I was and I had to remind myself that I hate doing math

oh the inside of the shape goes golden too. φ can leave the lines and float inside it when it does, which is interesting

I’m curious if Euler told Fibonacci about Orange at all, I can’t tell if φ is messing with Orange for fun or showing him who’s boss or just over-excited and going on a visual info-dump about its interests

what

why is it dubstep

what is this thing supposed to be?! I don’t remember learning about anything like this in math

that spiral was so satisfying though. never would’ve expected it to be accompanied by a fade-out from dubstep

OH THEY’RE SYNCHING NOW. THEY’RE CHOREOGRAPHING TO THE BEAT

I replayed the drawing of the glider several times, the sounds in that sequence just fell into place perfectly

math having a natural enemy is such a strange concept to entertain XD

the way the pentagon fractals when it breaks

Orange crafting again

ooh what do the other colors mean?

CONTINUITY?!

[the-z-part]

The 1973 masterpiece, Goncharov, has gone from celebrated, forgotten and now celebrated once again, suddenly emerging from obscurity. Given the cenematric importance of The Godfather in 1974, also directed by Scorsese, Jaws, directed by Spielberg in 1975, Rocky, directed by Avildsen in 1976 and Star Wars directed by Lucas in 1977, it is understandable that by 1978, there is almost no mention of Goncharov in cinema reviews or even textbooks. Only in academic journal articles and infrequent expansive reviews of gangster movies by critics who remember the excitement of Goncharov when it first appeared will you find any mention of the film. Yet, in the past year, perhaps because of increasing celebration of Scorsese’s oeuvre, has the title “Goncharov“ appeared again in print. 

One facet of the making of Goncharov that seems to have been missed, both in the 1970’s and now, is its intersection with mathematics. When scouting for film locations in Paris, Scorsese happened across a cafe near the École Normale Supérieure. Apparently, he ducked into the cafe to get out of an unseasonably cold weather. He struck up a conversation with a gentleman who happened to be none other than Nicolas Bourbaki, the mathematician who was a professor at ENS. Bourbaki is noted for his many textbooks starting in 1934 that were instrumental in the modernization of pure mathematics. The two became fast friends. Bourbaki and Scorsese took to having dinner and a long walk afterward every evening for the remainder of the three weeks Scorsese was in Paris. Bourbaki pointed out mathematics in nature on these evening walks, especially regarding proportions. 

For example, if one looks at branches on a tree, where the branches emerge from the trunk, it forms a spiral along the trunk, considering one branch to the next moving upward. This spiral will wrap around the trunk after a certain number of branches. The ratio of these numbers is consistent for a species of tree. An elm tree has a ratio of 1 to 2. That is, the spiral wraps once around the trunk every two branches. A beech tree has a ratio of 1 to 3. The willow tree has a ratio of 3 to 8. An almond tree has a ratio of 5 to 13. The oak tree, which is plentiful in Paris, has a ratio of 2 to 5. 

These spiral tree ratios come from the Fibonacci sequence, namely the infinite list that begins 1, 1, 2, 3, 5, 8, 13, 21, 54, … The first two numbers in the list are 1 and 1. After that, each new number in the list is the sum of the prior two values. The spiral ratios of trees are always made up of two numbers in the Fibonacci sequence that are two apart. So 1 and 2, then 1 and 3, then 2 and 5, then 3 and 8, etc. 

The seeds of a pine cone ascend in a spiral. The spiral goes in both directions. If one counts the number of pine seeds along a pine cone, the number of rows of seeds depends on which direction you are counting along the pine cone. Those two numbers will always be successive Fibonacci numbers. This is true also for the double spiral of seeds in a sunflower and the little square-ish units on the skin of a pineapple. 

In music, if the frequency of two notes is made up of successive Fibonacci numbers, then we have a special name for the interval. Two notes whose frequency is in a ratio of 1 to1 are in unison. In a ratio of 1 to 2 are an octave apart. In a ratio of 2 to 3 are a fifth apart. In a ratio of 3 to 5 are a fourth apart. The value of the ratio of successive terms in the Fibonacci sequence gets closer and closer to a particular value known as the golden ratio which is about 0.6180. 

Beyond discussing the golden ratio, Scorsese and Bourbaki also talked about other proportions like the silver ratio which plays a large role in Chinese and Japanese classical art. They discussed other constants, such as Euler’s number, e, which appears naturally in Calculus with simple questions about area as well as in finance with continuous compounding and in solutions to the motion of springs, electrical circuits and vibrations of molecules. They talked about the Euler’s constant, denoted with a Greek gamma, which appears in applications from information theory to celestial mechanics. Beyond importance in applications, Bourbaki impressed upon Scorsese that proportions are important to aesthetics. He apparently used the music of Bella Bartok as a motivating example. 

That meeting took place in 1972. Late that year, Scorsese had started filming in Rome and a few outlying villages that substituted as backdrops for Sicily. In November of that year, Scorsese invited Bourbaki to Rome to visit the set. Bourbaki arrived a few days later by train. The day after that, Bourbake had an uncredited cameo in the now-famous scene with Cybill Shepherd, Harvey Keitel and a very sharp knife.  

After filming was completed in early 1973, Scorsese went to work editing Goncharov. Most of the scenes from Paris ended up on the cutting room floor. The cameo of Bourbaki made it into the movie, thankfully. And Scorsese set the peak moments of action and the climax of the story using the golden ratio. The running time of the film is 212 minutes. The (emotional) climax is at 131 minutes. The ratio of 131 to 212 is 0.6179 and the golden ratio is 0.6180. The ticking clock image appears in the movie six times (“time is running out”). It appears in fact, every 34 minutes and 38 seconds with intense precision. The Euler constant, gamma, is about 0.5772. That portion of an hour is 34 minutes and 38 seconds. 

Almost as a send up to his friend, Scorsese has the silver bullion removed from the bank vault at 87 minutes and 49 seconds which represents the silver ratio (about 0.4143) to the whole film. At one over e, Euler’s number, (0.3679) which is about a third of the way through the film at 78 minutes, is when we meet the Accountant. Recall that Euler’s number is connected with finance, among other applications. These are not accidents. Scorsese experimented with these proportions with abandon in Goncharov. He apparently learned a great deal since many of these ratios appear in multiple ways in subsequent movies. 

Critic, Michael Koresky says of After Hours (Scorsese, 1985) “The nearly mathematical precision of After Hours was the result of a controlled yet rejuvenating movie shoot.” He is incorrect. The mathematical precision is not “nearly” anything. It is exact mathematical precision of proportions used in service to Scorsese’s vision. 

There is nothing in interviews or biographical material to suggest that Scorsese and Bourbaki met again after the making of Goncherov. Bourbaki’s work does not seem to be impacted by meeting Scorsese. For example, he does not write about films or proportions in service to narrative. Bourbaki’s output is prodigious through the 1950’s (except the war years) and he continued to publish intermittently after 1960. His most recent known work, as a centenarian for sure, was in 2016 on Algebraic Topology. The inscription of what is likely to be Bourbaki’s last work reads, “Pour Martin et nos promenades du soir.” (For Martin and our evening walks.) Perhaps Goncharov left its mark on Bourbaki, after all.

-- Sam Kaplan, Ph.D. in “Mathematical Sagacity” December 2022. 

[nick-shea]

Euler spiral

[theam-cjsw]

The AM: April 29, 2024

A meandering Monday morning, with a throwback to the one-and-only Moondog, a nod to late Quebecois singer-songwriter Jean-Pierre Ferland, and an hour of Cosmic American Music to wrap the show. That plus the usual strains of ambient electronics, meditative jazz, post-rock interludes, and other songs to start your week on solid footing.

Soundcloud and Spotify streams are after the break, along with the full track list. Or you can stream it direct from CJSW, your favourite campus and community station.

Hour One

Bron Ut Små Vågor, Henrik von Euler • Små Vågor 6

Muted - Lehto Remix Tristan De Liège • Refractions (Remixed)

Kind of Light - Khotin Remix Bodywash • Single

Sing Together Salamanda • Single

Fly! Little Black Thing Yu Ching • The Crystal Hum

Nautical Miles Castle If • From the Sea

Mirror Images Bristol Manor • A Distant Urban Forest

Transpubliction My Kill Jack's On • 4 In '84

Night Sculpture Forest Swords • Bolted

Pools Steve Lyman, featuring Ben Lukas Boysen • Spiral

Hour Two:

We’ll live through the long, long days and the longnights (And when our last hour comes we’ll go quietly) Eiko Ishibashi • Drive My Car OST

Portrait of the Artist as a Thursday Charles Spearin • My City Of Starlings

Afternoon Moon Pie Torngat • La Petite Nicole

Apple Tree Hintermass • The Apple Tree

Cuckoo Hill Ellis Island Sound • The Good Seed

What’s the Most Exciting Thing Moondog • Moondog 2

Wine, Women and Song Moondog • Moondog 2

Love is Blue Paul Mauriat • Single

Le chat du café des artistes Jean-Pierre Ferland • Jaune

ESGGallin Samantha Savage Smith • Fake Nice

Disarray Nolan Potter • The Perils of Being Trapped Inside a Head

Pink Lite Ryan Bourne • Plant City

Hour Three:

One Step Forward, Two Steps Back Chet Sounds • Changes Happen To Everyone, Everywhere

All the Time Ghost Woman • Ghost Woman

Little More Time ROY • Spoons for the World

Strange Insistence Gun Outfit • Out of Range

Cowgirl Ora Cogan • Formless

Doggie Paddlin’ Thru the Cosmic Consciousness Psychic Temple • Doggie Paddlin’ Thru the Cosmic Consciousness

The sharp smell of cedar Field Works • Cedars

Worship the Sun (Not the Golden Boy) Fiver • Audible Songs from Rockwood

Sunrise John Hulbert • Opus III

Sunlight, Good Light Clinton St. John • The Minor Arkhana

[mathsciencetuition]

What Prime Numbers Mean in Nature, Popular Culture, & the Internet

Prime numbers have captivated the imagination of mathematicians and scientists for centuries. These enigmatic numbers are not only essential in the field of mathematics but also find profound significance in various aspects of nature, popular culture, and the internet. At their core, prime numbers are integers greater than one that can only be divided by one and themselves without resulting in a fractional or decimal value. To explore the captivating world of prime numbers and excel in mathematics, students can benefit from specialized Maths tuition, such as that provided by the Miracle Learning Centre in Singapore. In this exploration, we will delve into the importance of prime numbers, uncovering their presence in the natural world, historical significance, cultural influences, and their ever-expanding role in the vast realm of the internet.

What are Prime Numbers?

Prime numbers are the fundamental building blocks of the natural numbers, and they play a crucial role in number theory and other branches of mathematics. An example of a prime number is 5, which can only be divided by 1 and 5, yielding a whole number quotient. In contrast, numbers like 4 or 6 can be divided by factors other than 1 and themselves, making them composite numbers.

The significance of prime numbers lies in their unique properties. They are infinite, and the distribution of prime numbers is a fascinating topic that has puzzled mathematicians for centuries. The prime number theorem, formulated in the 19th century, provides insights into how prime numbers become less frequent as numbers grow larger, but their irregular distribution defies any predictable pattern.

Importance of Prime Numbers:

Prime numbers hold immense significance in various aspects of mathematics and beyond. They serve as the foundational building blocks of number theory, cryptography, and other mathematical applications. Understanding prime numbers is essential for exploring and unlocking the mysteries of number patterns and mathematical relationships.

In addition to their mathematical importance, prime numbers have fascinating real-world applications. They play a vital role in encryption algorithms, ensuring secure communication and protecting sensitive data in the digital age. The study of prime numbers has also revealed their presence in nature, cultural expressions, and the internet, further highlighting their pervasive influence across diverse domains. The enigmatic nature of prime numbers continues to captivate mathematicians, scientists, and artists, making them a subject of endless exploration and fascination.

Prime Numbers in Nature:

Intriguingly, prime numbers also manifest in various aspects of the natural world. Some cicadas have life cycles that are prime number years long, which may be an adaptation to avoid synchronization with predators and increase their chances of survival. In the animal kingdom, prime numbers have been observed in the numbers of petals in certain flowers and the spirals of pinecones and sunflowers. These occurrences demonstrate the subtle yet remarkable presence of mathematical principles in the living world.

Prime Numbers in History:

Throughout history, prime numbers have captured the attention of mathematicians, leading to profound discoveries and breakthroughs. The ancient Greeks were among the first to explore prime numbers' properties, with Euclid proving that there are infinitely many primes around 300 BCE. The great mathematician Leonhard Euler made significant contributions to the study of primes in the 18th century, paving the way for further exploration in number theory.

Prime Numbers in Culture:

Beyond their mathematical significance, prime numbers have also permeated various cultural expressions. Writers, artists, musicians, and even filmmakers have drawn inspiration from these intriguing numbers. They have been used as symbols of mystery, uniqueness, and hidden patterns in literature, music, and visual arts. The fascination with prime numbers has made them a recurring motif in cultural works, connecting the abstract world of mathematics with the realm of human emotions and creativity.

Prime Numbers in the Internet:

In the digital age, prime numbers have gained newfound importance with the advent of cryptography and secure online communication. Large prime numbers serve as essential components of encryption algorithms, ensuring the confidentiality and integrity of data transmitted over the internet. Cryptographic systems, such as RSA encryption, rely on the computational difficulty of factoring the product of two large prime numbers, making it practically unfeasible for adversaries to break the code and access sensitive information.

Students can gain a deeper understanding of prime numbers and other fascinating mathematical concepts through specialized maths tuition. Enrolling in a Maths Tuition program, such as the one offered at Miracle Learning Centre in Singapore, can provide students with a structured and engaging learning environment. Expert maths teachers guide students through the intricacies of prime numbers, unraveling their significance in nature, culture, and the digital landscape.

Through interactive lessons, challenging exercises, and real-life examples, students can explore the enchanting world of mathematics, uncover interesting facts about prime numbers, and develop problem-solving skills that extend beyond the classroom. Maths tuition empowers students to embrace the beauty of numbers and inspires a lifelong appreciation for the wonders of mathematics.

Why Prime Numbers Matter?

The significance of prime numbers cannot be overstated, as they form the basis for essential applications in various fields. From secure communication and data protection to fundamental advances in number theory, prime numbers are at the core of numerous scientific and technological advancements. Understanding prime numbers can pave the way for groundbreaking discoveries and innovations, making them a vital area of study for mathematicians and researchers worldwide.

Understanding the profound importance of prime numbers can spark curiosity and interest in mathematics among students. Engaging in specialized maths tuition, such as the one provided by the Miracle Learning Centre in Singapore, can offer a unique opportunity for students to delve deep into the world of prime numbers under the guidance of expert Maths teachers in Singapore.

By enrolling in Maths Tuition in Singapore, students can unravel the enigmatic properties of prime numbers and their relevance in real-life applications. This learning experience empowers students to grasp complex concepts with ease, equipping them with invaluable mathematical skills and paving the way for future advancements and innovative solutions in the world of mathematics and beyond.

Some Interesting Facts about Prime Numbers:

The largest known prime number, as of my last update, is 2^82,589,933 - 1, a number with an astonishing 24,862,048 digits.

Prime numbers have fascinated mathematicians throughout history, with famous mathematicians like Gauss, Euler, and Ramanujan making significant contributions to the study of primes.

The twin prime conjecture proposes that there are infinitely many pairs of prime numbers that differ by 2, such as 3 and 5, 11 and 13, and so on.

The distribution of prime numbers follows no discernible pattern, making their occurrence appear random, yet their properties exhibit intriguing regularities.

Goldbach's conjecture, one of the oldest unsolved problems in number theory, states that every even integer greater than 2 can be expressed as the sum of two prime numbers.

Conclusion:

In conclusion, prime numbers hold a position of utmost importance in the realms of mathematics, nature, culture, and the internet. They form the backbone of number theory, cryptography, and various mathematical applications. Their presence in nature showcases the hidden mathematical order in the living world, while their influence on culture highlights the bridge between abstract concepts and human expression. Moreover, their role in securing internet communication has become indispensable in the digital age.

For those looking to grasp the true essence of prime numbers and their myriad applications, embarking on a journey of learning is essential. Students seeking to unravel the mysteries of mathematics can benefit from specialized maths tuition, such as that offered by the Miracle Learning Centre in Singapore. With the guidance of dedicated math teachers, students can explore the fascinating world of prime numbers and pave the way for future discoveries in this captivating field.

[newestmusic]

Listen/purchase: Halt! / Euler Spiral by Astral Quartet

[new-arcade]

Prompts: Owl, Swan, Lighthouse, Spiral/Circular/Confusion

I created a game where you are a bird flying around a lighthouse in the middle of the sea and try to catch fish by flying close to them.

I interpreted the prompts as birds-eye and the main player bird, the lighthouse as an element in the game and the circular movement of spiral prompt. The euler angle rotation is mapped to the vertical movement.

-Rama Deshpande

[art-of-mathematics]

Recursive brain noodle soup - Parody of paradoxes unrevels its transcending character - Uncoiling the contradictions in a dancing pattern

[This post is rather incomprehensible and incoherent gibberish, but I want to share these fragments of trains of thought nevertheless.]

Truth is neither day nor night. Truth is how day and night are like sine and cosine of the unit helix. Truth is a principle of interactions of all components in a data set. - Welcome to the analogy rollercoaster in the association weaving mill. - Enjoy the mental fuckery. And dont forget to have some fun along the way of getting your brain turned into semi-slimy noodle soup. Remember: "Creativity is intelligence having fun", as Carl Sagan once said it.

- - -- ---

Some days ago I bought this nice orange/yellow hoodie.

I like the "heterogeneous distribution" of the orange and yellow.

Today I attached some patches:

a patch with Euler"s identity/equation

a patch with an ironic dictionary-like explanation of the word Recursion (in german) [Recursion (Latin) see: Recursion]

a patch with the Fibonacci spiral, a patch with the flower of life pattern, a patch with a dodecahedron

AND - At the hood I attached a patch with some lyrics by the somewhat pagan-ish/shamanic-like band Heilung. [It is from their first live album LIFA (Opening ceremony) - Beware that the music by this band might have trance-inducing effects.] - This quote might appear like having a huge religious character. Yet, I don"t interpret it as such. It is an additional interesting plot twist to interpreting all the patches I have attached on that hoodie.

The lyrics on that patch:

"Remember that we all are brothers, All people, And beasts and trees, And stone and wind, We all descend from the one great being, That was always there, Before people lived and named it, Before the first seed sprouted."

How I interpret these lyrics: I ignore all details and think about the main principle(s) stated. According to that method, I translate these lyrics to:

"We all stem from the same. (->Darwin"s evolution ) We are all connected by the same rules of nature, and we all emerged by the same rules of nature. (This is difficult to explain and summarize... in short, I refer to interactions in dynamical systems and emergence. Rules of nature include laws of physics and all other sciences.

These laws of nature could be imagined as guiding principles for the metaphorical game of life, like a vague instruction how life should uncoil itself. )

These rules of nature already existed before we could discover them (alias "before we named them")"

[arbieroo-art]

Irrational Euler spirals.

[low-sugar-eye-candy]

Wait for it... 

[enby-axels]

Rika Kihira - 2020 Nationals - Baby, God Bless You (x)

[ryanlj]

Euler Spiral Earth