Water and Music

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It is five in the morning. I stand on a beach and gather light in my yes and my mind. It is not a normal beach but rather one with a bed of small round stones. They have no color yet, but promise to gather some as the sun rises.

There is a hint of pink on the horizon but no real light. The stars remain visible above the sea. Among them is the phenomenon of the Milky Way. It rises from the sea; upward and to the right, continuing towards the heavens until it disappears. There is not a clear demarcation of its trajectory. The lower parts of the Milky Way appear as stars – – although a bit muted. As my eyes follow its path the stars join together and lose their granularity; they become a giant cloud. This cloud has upper and lower limits which define it

There is a cliff to my left. It also, like the beach stones, has little color. A minute passes and some light tends to make the cliff a brownish hue. Yes, I can now clearly see its outline. It is not a normal flat cliff but rather one whose geological lineage have given it character. There are fault lines rising from left to right; almost following the path of the Milky Way. The fault lines are irregular, no two separated by the same distance although following the same path. There are a few outcroppings that cantilever out into the night sky.

Out, sitting in the ocean, with its toes on the beach and its heals buried in the water, is a tall rock island. Its birth mother appears to be the cliff. They have the same fault lines and color. Mother and child are separated by eroded rocks that appear to remain behind as the result of the natal event.

There are beach stones that lay beneath my feet, and extend to the area between the cliff and the island. These stones seem to also have been left behind as a result of the birth of the island. However, these have been smoothed by Father Ocean who has given them a personality due to a continuum of waves of advice.

In the distance and to my right a small island appears in the muted sun. The pink on the horizon has turned to white and a thin layer of blue exists between this and the dark sky. The stars within this blue band have disappeared; yet my heart can still feel the Milky Way and my mind contemplates it.

The sea becomes silver as the sun rises farther. The islands and cliff in front of me remain dark and foreboding. I wait for the remainder of the musical and am not disappointed. The small rounded stones appear to gain in color and harmony. I wonder and this wonderment leads me to the memory of a visit I once paid to a mountain stream.

I had a lot of leisure time then. My weekly income from industry gave me time to do the things I always wanted to. One of my desires was to own my own automobile so that I could visit the countryside.

And I did.

I drove for miles and enjoyed every month of it. Some of the roads were unique. One was along an old canal bed. Another went up and down like a roller-coaster (but, of course, not as steep). My favorite mountain road was helical shaped that went around and around as well as up. I enjoyed the feeling of “Déjà Vu” each time I made a left hand turn (which was continuously).

But none of my trips would be as memorable as the one I was on at that moment. It was a two lane macadam road that followed a beautiful mountain stream. I saw an interesting dirt road to my right. A rickety bridge allowed me to cross the stream. The steep mountains closed in on each side as I followed my hood ornament.

Suddenly the mountains parted and I found myself at the edge of a large flat area. It must have been five hundred acres in size and as flat as a postage stamp. The whole plain was covered in grass. It was not normal grass; it was blue. Not the blue of oceans, nor turquoise blue, or the blue color of melancholy. It was not the blue of azure, nor sapphire, nor peacock blue nor the blue of despair. It was the blue of amethyst.

It had the same quality of an amethyst gem; translucent. I departed my car and studied the grass. When I stood up it looked like a dark hue of blue. When I lay down and looked across the top of the grass it looked like the amethyst of an apothecary jar. In either case it was a mesmerizing experience.

I removed my shoes and socks to enjoy the softness of this blue grass. I walked around the amethyst plain for quite some time before I noticed a stream along one side. I ventured down the bank and into the water. The stones on the bottom were all the same size; about the size of an egg. Yet they were rather flat and pleasant to walk on. The stones appeared to have spent quite a bit of time in a giant lapidary tumbler; they were polished.

I reached down and picked one up. It was onyx black with ivory colored large spots here and there. The spots were not clearly demarcated as on a polka-dot dress. They were more like the penumbraic spots on a brook trout.

“PUT ME DOWN” cried a voice from seemingly nowhere.
I looked around but there was no one there.
“Put me down before I die” cried out the voice again.
I scanned the blue grass but saw no one.
“Please, I beg you, put me back in the water or I will expire” came the voice for the third time.

I perceived that the voice might be coming from the stone. I carefully lowered my hand until the stone was under water. Nothing happened. No voice, no movement, no sign of life. After a minute or so I decided that I must have had a day dream in this strange place of amethyst grass and shiny stones with soft ivory spots. I lifted my hand out of the water so that I could once more inspect this strange stone.

“Thanks for putting me back under water” said the stone.
Without thinking I responded “Why sure! Sorry about not listening the first time.”
“Put me back in so I can take a breath” pleaded the stone.
I did as I was asked and then lifted the stone up again. “Are you a real stone or a turtle or a fish?” I asked.
“A stone” said the stone. “Now put me back.”

And so it went for the remainder of the next hour. I dipped the stone in and out of the water while trying to complete a conversation with it.

Finally the stone told me “Walk downstream and you will find a waterfall.”
I followed the stone’s instructions while holding him under water. When I reached the waterfall I lifted the stone up again to ask for farther instructions. The stone said “Hold me under the waterfall. That way I can get enough water to breath and enough air to talk.”
I followed instructions and soon realized why I had heard gurgling sounds at all the waterfalls I had ever visited. It was the sound of stones talking to each other. I wished that I had paid more attention to stones long before this.

“I can now complete what I wished I could have told someone years ago” said the stone. “It is a long story so feel free to ask questions; if you must.
The stone continued “This is the story of a blue grass festival and us who were once called caterwaulers.

“Eons ago in this flat amethyst plain,
Music existed whether in sun or rain,
It echoed on the mountains ag’ane and ag’ane,
While the red summer hawk was carnivorously preyin’.

People came from a’near and afar,
Some by bus, but mostly by car,
Two people brought elixir in a stone jar,
A wanna-be king and a Tennesee tzar.

So the blue-grass music played day and night,
While the clouds rolled by, dark or bright,
Eventually the moon played fiddle (out of sight),
And the night owl prowled in quiet stealthy flight.

The stream rolled by, oblivious of the noise,
Created by those grown-up country girls and boys,
With their banjos, harmonicas, and musical toys,
The audience had elixir and hookahs (their make believe joys).

Summer hawk and night owl with eyes e’spyin’,
Swooping talons exposed, in unison flyin’,
Picked up the singers and musicians (all cryin’),
Each and every one thought they were dyin’,

The winged ones deposited them in mountain stream,
And the caterwauling ebbed to a gurgling scream,
I was involved in that nightmarish dream,
For I was a caterwauler, or so it would seem.

Over the eons minerals invaded our bones,
And turned us into penumbrious stones,
Our caterwaulings are now gurgling moans,
The screeching songs melted into melodious tones.”

The stone then became quiet except for intermittent sobbing. Although I had previously intended to ask questions I avoided hearing any more of the sad story. I carefully placed the stone on the bottom of the stream and fled.

I never heard the voice of the caterwauler again.

©  2017        Copyright Waldo J. Tomosky

Mathematics; through the ages: Post 24

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Hipparchus of Nicaea (Ἵππαρχος, c. 190 – 120 BCE)

Hipparchus of Nicaea was a Greek astronomer and mathematicians, and one of the greatest astronomers of antiquity.

Hipparchus made detailed observations of the night sky and created the first comprehensive star catalog in the western world. He is considered the father of trigonometry: he constructed trigonometric tables and used these to reliably predict solar eclipses. He also invented the astrolabe and solved different problems in spherical trigonometry.

Spherical trigonometry is a branch of trigonometry that deals with the relationships and measurements of triangles on the surface of a sphere. It is particularly important in astronomy and navigation, as it is used to solve problems related to the positions of celestial bodies, as well as determining locations on the Earth’s surface using celestial observations. Hipparchus played a crucial role in developing and systematizing the principles of spherical trigonometry.

Hipparchus’s work in spherical trigonometry involved defining the essential trigonometric functions for use on the celestial sphere, just as ordinary trigonometry deals with angles and triangles in a flat plane. He introduced concepts such as the chord length, which is a measure of the arc distance between two points on the celestial sphere, and the versine, which is the complement of the cosine in spherical trigonometry. Hipparchus also developed the fundamental relationships and theorems of spherical trigonometry, including the law of sines and the law of cosines for spherical triangles.

One of Hipparchus’s most notable achievements was the development of a comprehensive star catalog, which included the positions and brightness of over 1,000 stars. He also made precise measurements of the positions of the Sun and the Moon, allowing for more accurate predictions of lunar and solar eclipses. These observations and measurements were essential for his work in spherical trigonometry, as they provided the data needed to understand and model the motions of celestial bodies.

Hipparchus’s work laid the foundation for later astronomers and mathematicians, and his contributions to spherical trigonometry significantly influenced the development of astronomy and navigation. While many of his original writings have not survived, his ideas and methods have been transmitted through the works of later scholars, such as Ptolemy, who expanded upon Hipparchus’s work in their own astronomical treatises.

The end of Mathematics Through the Ages

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Mathematics; through the ages: Post 23

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Suàn shù shū

The Suàn shù shū (筭數書), which means Book on Numbers and Computation, is one of the oldest mathematical manuscripts from China. It was written around 200 BCE and consists of 200 strips of bamboo.

There are 69 problems, each with a solution, covering topics like arithmetic, fractions, integer factorization, geometric sequences, inverse proportions, unit conversion, and error handling. Geometry problems show how to find the area of circles and rectangles, as well as the volume of three-dimensional solids, while assuming that π=3.

Throughout Chinese history, there have been various suàn shù shū written for different purposes and audiences. These texts typically cover fundamental mathematical concepts, including basic arithmetic operations, number theory, algebra, geometry, and various problem-solving techniques. They are intended to teach people how to perform calculations and solve mathematical problems.

One of the most famous suàn shù shū in Chinese history is the “Nine Chapters on the Mathematical Art” (Jiuzhang Suanshu, 九章算术), which dates back to ancient China (circa 2nd century BCE). This text is one of the earliest known mathematical treatises in Chinese history and covers a wide range of mathematical topics, from simple arithmetic to more advanced algebraic methods. It has had a profound influence on the development of Chinese mathematics and served as a foundation for the study of mathematics in China for centuries.

Other notable suàn shù shū include the “Sea Mirror of Circle Measurements” (Yuanxi Suànfa, 圆 系 算 法) and the “Mathematical Manual in Nine Sections” (Jiuzhang Suànjing, 九章算经). These texts, along with many others, have contributed to the rich tradition of Chinese mathematics and its applications in various fields, including engineering, astronomy, and agriculture.

In summary, “Suàn shù shū” refers to arithmetic or mathematics books in the Chinese tradition, which have played a crucial role in teaching and advancing mathematical knowledge in China over the centuries.

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Mathematics; through the ages: Post 22

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Apollonius of Perga (c. 200 BCE)

Apollonius of Perga was a Greek mathematician and astronomer best known for his work on the four conic sections.

Apollonius of Perga, a Greek mathematician who lived around 262-190 BCE, made significant contributions to the study of conic sections. Conic sections are the geometric shapes formed by intersecting a plane with a cone. Apollonius is often referred to as the “Great Geometer” because of his extensive work on conic sections.

His major contributions in this field include:

  1. Classification of Conics: Apollonius provided a systematic classification of conic sections based on their defining characteristics. He distinguished between the different types of conics, which include the ellipse, parabola, and hyperbola. He described these shapes in terms of their mathematical properties, such as the focus, directrix, and eccentricity.
  2. Study of Ellipses: Apollonius made substantial progress in understanding ellipses. He introduced the concept of major and minor axes and studied properties related to the distances between points on the ellipse. His work laid the foundation for the development of analytical geometry, a field that would later become essential in the study of conic sections.
  3. Parabolas: Apollonius also extensively studied the parabola. He determined properties of the focus, directrix, and the vertex of the parabola. His work on parabolas is crucial for understanding their geometric and algebraic properties.
  4. Hyperbolas: Apollonius contributed to the understanding of hyperbolas as well. He examined the asymptotes and the relationships between the distances from the foci to points on the hyperbola. His work laid the groundwork for the later development of hyperbolas in mathematics.

Apollonius’ work on conic sections was groundbreaking and provided a strong foundation for the development of later mathematical theories. His treatise “Conics” consisted of eight books, some of which have been lost over time. Despite this loss, the surviving portions of his work significantly impacted the study of conic sections and geometry for centuries to come.

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Mathematics; through the ages: Post 21

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Eratosthenes of Cyrene (c. 276 – 195 BCE)

Eratosthenes of Cyrene, Libya, a Greek mathematician, geographer, astronomer, historian, and poet. He spent much of his life in Egypt, as head of the library of Alexandria. Among many other achievements, Eratosthenes calculated the circumference of the Earth, measured the tilt of the Earth’s axis of rotation, estimated the distance to the sun, and created some of the first maps of the world. He also invented the “Sieve of Eratosthenes”, an efficient way to calculate prime numbers.

Eratosthenes of Cyrene, developed a systematic procedure for finding prime numbers that is known as the sieve of Eratosthenes. Prime numbers are not divisible by an integer greater than 1, except themselves. 

What are the Sieve of Eratosthenes 1 to 100?

Prime Numbers — 1 to 100 — using ‘The Sieve of Eratosthenes ‘are;

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89, and 97.

There are 25 prime numbers between 1 to 100.

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Mathematics; through the ages: Post 20

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Archimedes Palimpsest

palimpsest is a scroll or parchment from which the text has been washed or scraped off so that it can be reused. This method was common in the Middle Ages – even for documents by brilliant scientists and mathematicians.

Archimedes of Syracuse lived in the 3rd Century BCE and was one of the greatest mathematicians in history. A Greek copy of some of his work, created around 1000 CE in Byzantium, was later overwritten by Christian monks in Palestine. More recently, forgers added pictures to increase the value of the documents.

In 1998, scientists started studying the Archimedes Palimpsest, and used X-rays, ultraviolet and infrared light to uncover the hidden original text.

The Archimedes Palimpsest contains unique copies of important works by Archimedes, such as “The Method of Mechanical Theorems” and “Stomachion,” as well as other mathematical texts. These texts provide insights into Archimedes’ mathematical methods and contributions to geometry and calculus. The palimpsest also contains other Greek texts from various authors, providing valuable information about the intellectual and scientific culture of the time when it was originally created.

The recovery and study of the Archimedes Palimpsest have significantly enriched our understanding of Archimedes’ work and his contributions to mathematics and science. It’s a remarkable example of how modern technology and meticulous efforts by scholars have allowed us to uncover lost or obscured knowledge from the past.

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Mathematics; through the ages: Post 19

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Archimedes (c. 287 – 212 BCE)

Archimedes was an ancient Greek scientist and engineer, and one of the greatest mathematicians of all time. He discovered many concepts of calculus and worked in geometry, analysis and mechanics.

While taking a bath, Archimedes discovered a way to determine the volume of irregular objects using the amount of water they displaced when submerged. He was so excited by this discovery that he ran out on the street, still undressed, yelling “Eureka!” (Greek for “I have found it!”).

As an engineer, he built ingenious defense machines during the siege of his home city Syracuse in Sicily. After two years, the Romans finally managed to enter, and Archimedes was killed. His last words were “Do not disturb my circles” – which he was studying at the time.

The Archimedean solids, also known as the Archimedean polyhedra, are a group of convex polyhedra with identical vertices and faces that are regular polygons. There are a total of 13 Archimedean solids:

  1. Truncated Tetrahedron
  2. Truncated Cube (also known as the Cuboctahedron)
  3. Truncated Octahedron
  4. Truncated Dodecahedron
  5. Truncated Icosahedron
  6. Rhombicuboctahedron
  7. Great Rhombicuboctahedron
  8. Truncated Cuboctahedron
  9. Snub Cube
  10. Icosidodecahedron
  11. Truncated Dodecahedron
  12. Truncated Icosidodecahedron
  13. Snub Dodecahedron

Each of these Archimedean solids has a unique set of characteristics and properties. They are defined by specific combinations of regular polygons as their faces, and they can be a fascinating subject for geometry and mathematics enthusiasts.

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Mathematics; through the ages: Post 18

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Euclid’s Elements

Around 300 BCE, Euclid of Alexandria wrote The Elements, a collection of 13 books that contained mathematical definitions, postulates, theorems and proofs, and covering topics like geometry and number theory.

It is one of the most famous books ever written, and one of the most influential works in the history of mathematics. Copies were used as textbooks for thousands of years and studied all around the world, with thousands of new editions published.

No original copies of the Elements still exist today. This small papyrus fragment dates back to around 100 AD, and may be a part of the oldest existing copy of Euclid’s work. It is part of the Oxyrhynchus papyri, which were found in 1897 in an ancient rubbish dump in Egypt. The diagram shows the 5th proposition in book 2 of the Elements, a geometric version of the identity (x+y) X (x−y) = x2−y2

  1. Structure: The Elements is divided into thirteen books, each covering a specific area of geometry and mathematics. The books are organized as follows:
    • Books I-IV: Basic principles of plane geometry, including definitions, postulates, and common notions.
    • Books V-VI: Theory of proportions and similarity.
    • Book VII: Number theory, specifically focusing on prime numbers and divisibility.
    • Books VIII-IX: Geometric algebra, including the theory of proportionality.
    • Book X: Commensurable magnitudes and incommensurable magnitudes.
    • Books XI-XIII: Solid geometry, including the study of three-dimensional figures.
  2. Axiomatic System: Euclid’s Elements is notable for its use of a rigorous axiomatic system. It starts with a set of definitions, postulates (also known as axioms), and common notions that serve as the foundation for all the subsequent theorems and proofs. This approach helped establish the logical underpinnings of geometry and is considered a cornerstone of modern mathematics.

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Mathematics; through the ages: Post 17

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Bamboo Multiplication Table

Here you can see a set of 21 Bamboo Strips that were created around 2300 years ago in China. When arranged correctly, they form a multiplication table in base 10, written in ancient Chinese calligraphy.

While earlier civilizations like the Babylonians created multiplications tables in base 60, this is by far the oldest known decimal multiplication table – and it looks very similar to what we still use today.

These bamboo strips, like other historical texts, were discovered in various archaeological sites in China. Their discovery has contributed to our understanding of how mathematical concepts were developed and used in ancient times.

Scholars and historians have studied these multiplication tables to gain insights into the mathematical practices of ancient China. They provide evidence of advanced mathematical thinking in the region and help trace the historical development of mathematical concepts.

These bamboo strips with base-10 multiplication tables are significant in showcasing the mathematical knowledge and practices of ancient China. They offer a unique perspective on how arithmetic and multiplication were understood and applied in early Chinese civilization.

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Mathematics; through the ages: Post 16

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Pingala (पिङ्गल)

Below is Pingala’s creation of Pascal’s Triangle before Pascal.

Did Pingala discover the Fibonacci sequence?

Where was the Fibonacci sequence discovered? While Fibonacci himself did not discover Fibonacci numbers (they were named after him), he did use them in Liber Abaci. The numbers originate back to ancient India and are credited to the Indian Mathematician Pingala.

The pattern hides a powerful secret: If you divide each number in the sequence by its predecessor (except for 1 divided by 0), then as you move toward higher numbers, the result converges on the constant phi, or approximately 1.61803, otherwise known as the golden ratio.

The following link shows several items of nature that follow the golden ratio; including humans, trees, flowers, shells (such as the nautilus) and even galaxies.

https://www.mathnasium.com/blog/14-interesting-examples-of-the-golden-ratio-in-nature#:~:text=For%20example%2C%20the%20measurement%20from,%2C%20ants%2C%20and%20honey%20bees.

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Mathematics; through the ages: Post 15

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Euclid of Alexandria (Εὐκλείδης, around 300 BCE)

Euclid of Alexandria  was a Greek mathematician and is often called the father of geometry. His book The Elements first introduced Euclidean geometry, defines its five axioms, and contains many important proofs in geometry and number theory – including that there are infinitely many prime numbers. It is one of the most influential books ever published, and was used as textbook in mathematics until the 19th century.

Euclid taught mathematics in Alexandria, but not much else is known about his life.

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