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Old November 23rd, 2022 #1
jagd messer
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Default Mathematics


November 23, 2022


Fibonacci Day - Wednesday, November 23, 2022 Items & Things Mathematics There are sequences that appear in nature time and time again, ones that seem to define the very basis of our reality and coordinate how everything comes together. One of these numbers is the Fibonacci sequence, and it can be found in the most surprising of places.



Also known as Leonardo of Pisa and Leonardo Fibonacci, Leonardo Bonacci invented a pattern of counting that continues to influence Maths and technology today. The pattern is made up of numbers that sum the previous two numbers before them — 1, 1, 2, 3, 5, 8, 13 — and so on. The sequence is used in computing, stock trading, and architecture and design.

Once we discovered the sequence, it started showing up everywhere. Nature is full of Fibonacci patterns, from DNA to hurricanes, leading some to dub the Fibonacci sequence “nature’s secret code.”

http://Fibonacci Day - November 23, ...f The Holidays



HISTORY OF FIBONACCI DAY



One of the most important mathematicians of the Middle Ages, Leonardo Bonacci — later known as Fibonacci, “the son of” Bonacci — invented a sequence of numbers that shows up constantly in nature, physics, and design.

Born to an Italian merchant, the young Leonardo traveled to North Africa with his father, where he was exposed to the Hindu-Arabic numeral system. The system, which includes zero and limits itself to 10 symbols, is much more agile and flexible compared to the unwieldy Roman numeral system. In 1202, Fibonacci published “Liber Abaci”, introducing Europe to the Hindu-Arabic system and his now-famous sequence.

Starting with 1, 1, 2, 3, 5, the Fibonacci sequence is created by adding up the two previous numbers to get the next one. Fibonacci’s original example for his sequence pondered the population growth of rabbits. If starting with one pair, and each month that pair bears a new pair, the number of rabbits will grow at a rate consistent with his pattern of numbers.

The Golden Ratio, a proportion associated with the Fibonacci sequence and also frequently found in nature, is roughly 1 to 1.6. This ratio shows up in the branching patterns of trees, the distribution of seeds in berries, the spiral arms of galaxies, and many more natural and human-engineered things.

Fibonacci Day celebrates this important mathematician and gives us an opportunity to marvel at the way math pervades everything around us. The Fibonacci sequence can be used to calculate the proportions of countless things on Earth and beyond, such as animals, plants, weather patterns, and even galaxies. Pause to observe your surroundings and you’ll start to notice the familiar spiral all around you.



HOW TO CELEBRATE FIBONACCI DAY


Learn the sequence.

How far in the sequence can you go? Remember, each number is the sum of the two numbers before it. Try reciting the sequence and see how far you get.

FIBONACCI DAY - November 23, 2022 - National Today
 
Old November 23rd, 2022 #2
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Old January 6th, 2023 #3
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Default Pi irrational number, non terminating decimal, so cannot be written as p/q


Pi Day | Celebrate Mathematics on March 14th



https://www.piday.org
Pi Day is celebrated on March 14th (3/14) around the world. Pi (Greek letter “ π ”) is the symbol used in mathematics to represent a constant — the ratio of the circumference of a circle to the diameter…


Learn about Pi / Pi Day



What is Pi
What is Pi? Probably no symbol in Mathematics has evoked so much . . .



Pi Symbol

Pi Symbol As a Maths student, you learned that pi is the value that is . . .


One Million Digits of Pi
One Million Digits of Pi The First Ten Digits of Pi (π) are 3.1415926535. The First Million Are . . .
 
Old January 16th, 2023 #4
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Default Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia.


Pascal's Triangle - Math is Fun

https://www.mathsisfun.com/pascals-triangle.html



Pascal's Triangle shows us how many ways heads and tails can combine. This can then show us the probability of any combination. For example, if you toss a coin three times, there is only one combination that will give three heads …


Patterns Within the Triangle:



pascals triangle 1s, counting, triangular

Diagonals:



The first diagonal is, of course, just "1"s

The next diagonal has the Counting Numbers (1,2,3, etc).

The third diagonal has the triangular numbers

(The fourth diagonal, not highlighted, has the tetrahedral numbers.)

Horizontal Sums:


What do you notice about the horizontal sums?

Is there a pattern?

They double each time (powers of 2).

Exponents of 11:

Each line is also the powers (exponents) of 11:

110=1 (the first line is just a "1")
111=11 (the second line is "1" and "1")
112=121 (the third line is "1", "2", "1")
etc!

Squares:

For the second diagonal, the square of a number is equal to the sum of the numbers next to it and below both of those.

Fibonacci Sequence:

Try this: make a pattern by going up and then along, then add up the values (as illustrated) ... you will get the Fibonacci Sequence.

(The Fibonacci Sequence starts "0, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc)

Polynomials:

Pascal's Triangle also shows us the coefficients in binomial expansion
 
Old January 28th, 2023 #5
jagd messer
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Default Fractions that are Recurring decimals

Fractions that are Recurring decimals:

SEVENTHS


1/7, 2/7, 3/7, 4/7, 5/7, 6/7

1/7 is 1 ÷7 1.000000 ÷7= 0.142857142857 which is 0.1°42857° written with recurring dots at the beginning and end of the strip that recurs. Easy to remember 14 28 57

2/7 is 2 ÷7,2.000000 ÷7=0.285714285714 which is 0.2°85714°

3/7 is 0.4°28571°

4/7 is 0.5°71428°

5/7 is 0.7°14285°

6/7 is 0.857142

There is a strip of six digits, divide the numerator by the denominator and thats where you you start on the strip going right unto you have six digits.


Other Fractions that are Recurring decimals:


1/3, 2/3

1/6, 5/6

1/7, 2/7, 3/7, 4/7, 5/7, 6/7

1/9, 2/9, 3/9, 4/9, etc.

1/11, 2/11, etc.

1/12, 5/12, 7/12, etc.

1/13, 2/13, 3/13 etc. list these.

1/19, etc. Using paper find how big is this recurring strip is? Write out the 19 times table: 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, etc. to 342

List 1/19, 2/19, 3/19, etc to 18/19.

1/19 0.0°526315789478947368421°05 – 22 digits

2/19 2.00. . . ÷by 19 where do you start on the 22 digit strip

3/ 19 continue

. . .

1/23, 2/23, etc.

Couldn't find the recurring dot on this laptop had to use degree symbol °
 
Old April 27th, 2023 #6
jagd messer
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Default What Is So Special About The Number 1.61803?



PHI(φ) is an irrational, non-terminating number as PI(π), but its significance is far more than PI(π) ;

Π = 3.14159265359…(pi)

Φ = 1.61803398874…(phi)

The Golden Ratio (phi = φ) is often called The Most Beautiful Number In The Universe.

The reason φ is so extraordinary is because it can be visualized almost everywhere, starting from geometry to the human body itself!

The Renaissance Artists called this “The Divine Proportion” or “The Golden Ratio”.

PHI(φ) can be seen appearing in the following ways:

1. Fibonacci Series



0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

This series was developed by an Italian mathematician known as Leonardo Fibonacci. Other than the fact that each term is the sum of its two preceding consecutive terms, it can also be seen that if we divide a term greater than 2 by a term preceding it that the ratio always tends to 1.618…!

And if we continue this division after the 13th term we will always get a fixed number = 1.618

Example;

89/55 = 1.618

144/89 = 1.618

233/144 = 1.618

377/233 = 1.618

610/377 = 1.618

987/610 = 1.618

So on…!!!

2. The Human Body



For instance, if you divide the length from your head to toe by the length from your bellybutton to toe you will find the answer tending to φ.
Now, divide the length from your shoulder to the tip of the index finger by the length from your elbow to the wrist (of the same arm) and you’ll get φ again..!!
Divide the length from the top of the head to the shoulder by the length from your top of the head to your chin, φ again!
Top of your head to belly button by the length between you head and shoulder…..BANG….φ again!!!
Distance between your bellybutton and the knee, by the distance between knee and the bottom of the foot….φ again!
Now divide the length of your face to the width of the face……BAAM…φ again!!
Width of your two upper teeth to that of its height, and you’ll get φ again!
Lips to eyebrow divided by the length of the nose, φ again!


3. Plants



A sunflower grows in opposing spirals, the ratio of its rotation’s diameter to the next is 1.618…..i.e. φ again!
The ratio between the margin of a leaf to its veins(some plants) also gives φ.

4. DNA Of Organisms



DNA of the cell appears as a double-stranded helix referred to as B-DNA. This form of DNA has a two groove in its spirals, with a ratio of φ in the proportion of the major groove to the minor groove.
A cross-sectional view from the top of the DNA double helix forms a decagon. A decagon is actually two pentagons, with one rotated by 36 degrees from the other, so each spiral of the double helix must trace out the shape of a pentagon. The ratio of the diagonal of a pentagon to its side is φ to 1.

5. The Solar System



The average of the mean orbital distances of each successive planet in relation to the one before, tends to φ.
The Kepler’s Triangle(the triangle formed by utilizing the moon and the earth) is formed by a Pythagorean relation, in which the three sides of the right-angled triangle formed are always of this order:
Hypotenuse = φ

Perpendicular = √φ

Height = 1

If the rings of Saturn are closely looked at we will see that there is a ring that is quite denser than the other rings. Miraculously this inner ring exhibits the same golden section proportion as the brighter outer ring i.e. φ
Venus and the Earth are linked in an unusual relationship involving φ. If Mercury represents the basic unit of orbital distance and period in the solar system:
we find:

√Period of Venus * φ = Distance of the Earth

√2.5490 * 1.6180339 = 1.5966 * 1.6180339

= 2.5833 million kilometers

6. Art And Architecture



The Golden Ratio was probably most utilized by artists and architects while building their masterpieces. The following 5 pieces of work are specifically mentioned in the list as the golden ratio has been extensively used while creating them!

The Great Pyramid of Giza
Notre Dame
The Vitruvian Man
The Last Supper
The Parthenon

7. Music

[IMG]https://miro.medium.com/v2/resize:fit:720/format:webp/1*D2L-hhmPC8Kr3e9rsTj0Bg.jpeg[/IMG]

If we divide an octave by a perfect fifth, (13/20) = φ

If we divide a perfect fifth by an octave, (8/13) = φ

If we divide a perfect fourth by a major sixth, (6/10) = φ

And if we divide a major third by a perfect fifth, (5/8) = φ

Therefore we can see that φ is indeed a mystical number which can be visualized all around us.

And if we observe closely we can find its traces going back before humanity was even inhabiting earth, for example, the skin folds of extinct dinosaurs, rare ancient insect segmentation, and much beyond that.

***
Edit: I can see that some people are confused about the fact that how dividing 13 by 20 gives the answer as “phi”. So here’s the answer:

One of the very basic properties of “phi” is that its reciprocal is 1 less than itself(or o.618). You can either compare the numbers listed in the above quoted part of the article(about the octaves) to this value(0.618) or just flip the fraction to get the answer around 1.618.

Therefore you can use 1.618 and 0.618 interchangeably!


What Is So Special About The Number 1.61803? | by...

medium.com›@gautamnag279…so-special…number…61803…
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PHI(φ) is an irrational, non-terminating number as PI(π), but its significance is far more than PI(π) ; The reason φ is so extraordinary is because it can be visualized almost everywhere, starting…
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Old April 27th, 2023 #7
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I checked out a DVD from my library recently about fractal geometry and it was very interesting. A mathematician published a book about the subject in 1978 saying he had discovered these fractal patterns in many places in nature. This is the first time that this phenomenon had been given documentation. He showed that the fractal patterns diverge from the orginal and become smaller in the same form, such as with the branches of a tree.
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Old June 3rd, 2023 #8
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Old June 28th, 2023 #9
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Default Three Pythagoreans are talking about their life journeys.

Three Pythagoreans are talking about their life journeys.

Here is one for you to try:

Three Pythagoreans are talking about their life journeys.
The eldest said to the second oldest, “Fifteen years ago I was your age.”
The second turned to the youngest and said, “Fifteen years ago I was your age.”
The youngest then replied: “The square of my age plus the square of your age is equal to the square of our venerable elder’s age.”
How old is each Pythagorean?


Remember:
The Pythagorean (or Pythagoras') Theorem is the statement that the sum of (the areas of) the two small squares equals (the area of) the big one. In algebraic terms, a˛ + b˛ = h˛ where h is the hypotenuse while a and b are the legs of the triangle. The theorem is of fundamental importance in Euclidean Geometry where it serves as a basis for the definition of distance between two points.
 
Old July 6th, 2023 #10
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Default Three Pythagoreans are talking about their ages

PROBLEM:
Three Pythagoreans are talking about their life journeys. Three Pythagoreans are talking about their life journeys. The eldest said to the second oldest, “Fifteen years ago I was your age.”

The second turned to the youngest and said, “Fifteen years ago I was your age.”

The youngest then replied: “The square of my age plus the square of your age is equal to the square of our venerable elder’s age.”

How old is each Pythagorean?


SOLUTION:

Let x be the youngest so
x, x + 15 and x + 30

(x)˛, (x + 15)˛ , (x + 30)˛

x˛ + x˛ + 30x +225 = x˛ +60x +900

2x˛ -x˛ + 30x -60x + 225 -900 = 0

x˛ -30x – 675 = 0

(x + 15)(x -45) = 0

x= -15, x= 45

if x = 45, x+ 15 = 60 and x+30 = 75

(Pythagorean triple 3, 4 and 5) by 15.




or could have let x be the oldest so
x – 30, x -15 and x

then (x -30)˛ + (x - 15)˛ = x˛

x˛ - 60x + 900 + x˛ - 30x + 225 = x˛

x˛ - 90x + 1125 = 0

(x - 75)(x - 15) = 0

x = 75 or x = 15

If x=75 then ages are 75, 60 and 45.



One more:


 
Old July 30th, 2023 #11
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Default Tetrahedral Number Sequence - MATHS

Tetrahedral Numbers

A number is termed as a tetrahedral number if it can be represented as a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number is the sum of the first n triangular numbers.
The first ten tetrahedral numbers are:
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, …



Like stacks of cannonball.


Tetrahedral Number Sequence - MATHS



Polyhedron - Wikipedia
https://en.wikipedia.org/wiki/Polyhedron

In geometry, a polyhedron (plural polyhedra or polyhedrons; from Greek πολύ (poly-) 'many', and εδρον (-hedron) 'base, seat') is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on … See more


Regular polyhedra


The five convex examples have been known since antiquity and are called the Platonic solids. These are the triangular pyramid or tetrahedron, cube, octahedron, dodecahedron and icosahedron:

Polyhedrons: the names are tetrahedra, hexahedra (cube), octahedra (8-sided polyhedra), dodecahedra (12-sided polyhedra), and icosahedra (20-sided polyhedra).





 
Old July 30th, 2023 #12
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Default Tetrahedral Number Sequence

This is the Tetrahedral Number Sequence:

1, 4, 10, 20, 35, ...

We can understand it better when we think of a stack of marbles in the shape of a Tetrahedron.

Just count how many marbles are needed for a stack of a certain height.

For height=1 we only need 1 marble
For height=2, we need 4 marbles (1 at the top and 3 below)
For height=3 we need 10 marbles.
For height=4 we need 20 marbles.
For height=5 we need 35 marbles.
How many for height=6 ... ?



Tetrahedral Number Sequence

mathsisfun.com›tetrahedral-number.html

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Old July 30th, 2023 #13
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Default Perfect Numbers

Perfect numbers

Any positive integer that is equal to the sum of its distinct proper factors (factors other than the number itself).

Example: 6 (proper factors: 1, 2, 3) is a Perfect number because 1+2+3=6.



Example: 28 (proper factors: 1, 2, 4, 7, 14) is also a Perfect number, because 1+2+4+7+14=28.

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first few perfect numbers are 6, 28, 496 and 8128.

History

In about 300 BC Euclid showed that if 2p − 1 is prime then 2p−1(2p − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100.[2] In modern language, Nicomachus states without proof that every perfect number is of the form

2^{n}-1 is prime.[3] He seems to be unaware that n itself has to be prime. He also says (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6, 8, 6; but the sixth also ends in 6.) Philo of Alexandria in his first-century book "On the creation" mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen, and by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 14–19). St Augustine defines perfect numbers in City of God (Book XI, Chapter 30) in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician Ismail ibn Fallūs (1194–1252) mentioned the next three perfect numbers (33,550,336; 8,589,869,056; and 137,438,691,328) and listed a few more which are now known to be incorrect. The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician. In 1588, the Italian mathematician Pietro Cataldi identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8.

Perfect number - Wikipedia
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Old July 30th, 2023 #14
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Default The mystique of mathematics: 5 beautiful Math phenomena

Mathematics is visible everywhere in nature, even where we are not expecting it. It can help explain the way galaxies spiral, a seashell curves, patterns replicate, and rivers bend.


Fractals - patterns that repeat themselves on smaller scales - can be seen frequent

Even subjective emotions, like what we find beautiful, can have a mathematical explanation.

"Maths is not only seen as beautiful—beauty is also mathematical," says Dr. Thomas Britz, a lecturer in UNSW Science's School of Mathematics & Statistics. "The two are intertwined."

Dr. Britz works in combinatorics, a field focused on complex counting and puzzle solving. While combinatorics sits within pure mathematics, Dr. Britz has always been drawn to the philosophical questions about mathematics.

He also finds beauty in the mathematical process. "From a personal point of view, maths is just really fun to do. I've loved it ever since I was a little kid. "Sometimes, the beauty and enjoyment of maths is in the concepts, or in the results, or in the explanations. Other times, it's the thought processes that make your mind turn in nice ways, the emotions that you get, or just working in the flow—like getting lost in a good book."

Here, Dr. Britz shares some of his favorite connections between maths and beauty.

1. Symmetry—but with a touch of surprise


Symmetry is everywhere you look.

In 2018, Dr. Britz gave a TEDx talk on the Mathematics of Emotion, where he used recent studies on maths and emotions to touch on how maths might help explain emotions, like beauty. "Our brains reward us when we recognize patterns, whether this is seeing symmetry, organising parts of a whole, or puzzle-solving," he says. "When we spot something deviating from a pattern—when there's a touch of the unexpected—our brains reward us once again. We feel delight and excitement." For example, humans perceive symmetrical faces as beautiful. However, a feature that breaks up the symmetry in a small, interesting or surprising way—such as a beauty spot—adds to the beauty.

"This same idea can be seen in music," says Dr. Britz. "Patterned and ordered sounds with a touch of the unexpected can have added personality, charm and depth."

Many mathematical concepts exhibit a similar harmony between pattern and surprise, elegance and chaos, truth and mystery.

"The interwovenness of maths and beauty is itself beautiful to me,"
says Dr. Britz.


Each frond of a fern shoots off smaller versions of themselves. Sometimes, the fron

2. Fractals: infinite and ghostly

Fractals are self-referential patterns that repeat themselves, to some degree, on smaller scales. The closer you look, the more repetitions you will see—like the fronds and leaves of a fern.

"These repeating patterns are everywhere in nature," says Dr. Britz. "In snowflakes, river networks, flowers, trees, lightning strikes—even in our blood vessels."

Fractals in nature can often only replicate by several layers, but theoretic fractals can be infinite. Many computer-generated simulations have been created as models of infinite fractals.

"You can keep focusing on a fractal, but you'll never get to the end of it,"
says Dr. Britz.

"Fractals are infinitely deep. They are also infinitely ghostly.

"You might have a whole page full of fractals, but the total area that you've drawn is still zero, because it's just a bunch of infinite lines."



The Mandelbrot Set is arguably the most famous computer-generated fractal. Zooming in will reveal the exact same image on a smaller scale – a dizzying and hypnotic endless loop.

3. Pi: an unknowable truth

Pi (or 'π') is a number often first learned in high school geometry. In simplest terms, it is a number slightly more than 3.

Pi is mostly used when dealing with circles, such as calculating the circumference of a circle using only its diameter. The rule is that, for any circle, the distance around the edge is roughly 3.14 times the distance across the center of the circle.

But Pi is a lot more than this.

"When you look into other aspects of nature, you will suddenly find Pi everywhere," says Dr. Britz. "Not only is it linked to every circle, but Pi sometimes pops up in formulas that have nothing to do with circles, like in probability and calculus."

Despite being the most famous number (International Pi Day is held annually on 14 March, 3.14 in American dating), there is a lot of mystery around it.

"We know a lot about Pi, but we really don't know anything about Pi," says Dr. Britz.

"There's a beauty about it—a beautiful dichotomy or tension."



Pi is tied to ocean and sound waves through the Fourier series, a formula used in r

Pi is infinite and, by definition, unknowable. No pattern has yet been identified in its decimal points. It's understood that any combination of numbers, like your phone number or birthday, will appear in Pi somewhere (you can search this via an online lookup tool of the first 200 million digits).

We currently know 50 trillion digits of Pi, a record broken earlier this year. But, as we cannot calculate the exact value of Pi, we can never completely calculate the circumference or area of a circle—although we can get close.

"What's going on here?" says Dr. Britz. "What is it about this strange number that somehow ties all the circles of the world together?

"There's some underlying truth to Pi, but we don't understand it. This mystique makes it all the more beautiful."

4. A golden and ancient ratio

The Golden Ratio (or 'ϕ') is perhaps the most popular mathematical theorem for beauty. It's considered the most aesthetically pleasing way to proportion an object.

The ratio can be shortened, roughly, to 1.618. When presented geometrically, the ratio creates the Golden Rectangle or the Golden Spiral.

"Throughout history, the ratio was treated as a benchmark for the ideal form, whether in architecture, artwork, or the human body," says Dr. Britz. "It was called the "Divine Proportion."


The Golden Spiral is often used in photography to help photographers frame the image

"Many famous artworks, including those by Leonardo da Vinci, were based on this ratio."

The Golden Spiral is frequently used today, especially in art, design and photography. The center of the spiral can help artists frame image focal points in aesthetically pleasing ways.

5. A paradox closer to magic

The unknowable nature of maths can make it seem closer to magic.

A famous geometrical theorem called the Banach-Tarski paradox says that if you have a ball in 3-D space and split it into a few specific pieces, there is a way to reassemble the parts so that you create two balls.

"This is already interesting, but it gets even weirder," says Dr. Britz.

"When the two new balls are created, they will both be the same size as the first ball."

Mathematically speaking, this theorem works—it is possible to reassemble the pieces in a way that doubles the balls.


Duplicating balls is impossible - right?

"You can't do this in real life," says Dr. Britz. "But you can do it mathematically.

"That's sort of magic. That is magic."

Fractals, the Banach-Tarski paradox and Pi are just the surface of the mathematical concepts he finds beauty in.

"To experience many beautiful parts of maths, you need a lot of background knowledge," says Dr. Britz. "You need a lot of basic—and often very boring—training. It's a bit like doing a million push ups before playing a sport.

"But it is worth it. I hope that more people get to the fun bit of maths. There is so much more beauty to uncover."

Provided by University of New South Wales


The mystique of mathematics: 5 beautiful Math phenomena

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Old July 30th, 2023 #15
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Old July 30th, 2023 #16
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Quote:
Originally Posted by jagd messer View Post
I found them.
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Old November 3rd, 2023 #17
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Default sumerian sexagesimal system divide our clocks and circles

Why the hour’s dividers are called minutes and minute’s dividers are called seconds?


Perhaps some reader has ever asked about so seemingly irrelevant question as the origin of the names “minute” and “second”, the time’s dividers, but any knowledge is valuable.

History Begins at Sumer: Thirty-Nine "Firsts" in Recorded History (1956) is the title of a famous work of Noah Kramer published in the fifties of the last century. The Sumerians were pioneered on writing, astronomy, mathematics, etc.. Egyptians learned a lot from them and from both (Sumerians and their descendants in the area, Babylonians, Persians, etc.., and Egyptians) learned Greeks and other Mediterranean peoples.

In Sumer there is any indication of a numeral system with five as the base, quinary, undoubtedly related to the five fingers. They also used a dozenal (duodecimal) system with base twelve, base-12; (they had it pointing with the thumb three phalanges in each of the remaining four fingers); it is related to the twelve moons of the year … They used also a decimal system, denary ),base-10, associated with the ten hand’s fingers.

Interestingly also they used a sexagesimal system, (base 60) but we do not know exactly its origin. It is thought that this system facilitated the equivalence between the decimal and dozenal (duodecimal), since the divisors of 60 are 1, 2, 3, 4,5, 6, 10, 12, 20, 30, 60.

Well, from the dozenal (duodecimal) and sexagesimal systems, from the Sumerians approaches and Egyptian and Greek , it was established the division of time in hours, these in minutes, and these in seconds. And this is the system that we employed.

The same system is applied to the division of space.

So the current measures of the angles in degrees and minutes, this one of the clock face or timing measure, and this one of the earth orb globe with latitude and longitude coordinates, have their origin in the numeral system invented by the Sumerians and Babylonians 4,000 years ago.

Eratosthenes, Greek mathematician and astronomer (c. 276-194 BC) used the sexagesimal system to divide the circle into 60 parts or degrees (the Latin word gradus means step) and created the horizontal parallel lines from east to west to indicate the latitude.

A century later Hipparchus created a system of vertical lines going from north to south, dividing the sphere into 360 degrees.

At mid-second century A.D. Ptolemy developed the work of Hipparchus and divided the 360 degrees each one into 60 smaller parties in his work “Almagest”. These fractions are called "partes minutae primae", ie, "first small parts." Again he fractionated these first parts into other smaller 60, and he called these "partes minutae secundae ", ie "second small parts."

The first fraction, the "partes minutae primae" ended up being called "minutae", "small", from where we derive our word "minute".

The second fraction, "partes minutae secundae", "second small parts", ended up being called "secundae", and from this were derive the word "second".

Note: Ptolemy wrote his work in Greek and it is named Μαθηματικὴ Σύνταξις (Mathematike Syntax), Mathematical Composition. In Latin it was known as Syntaxis mathematica. It was then called 'Ἡ Μεγάλε Σύνταξις' He Megale Syntaxis, Great syntax, the Great Treatise.

The superlative of Greek adjective μεγάλε, megale, great, big, is μεγίστη, μέγιστος, megiste, megistos, (the greatest, the largest). The word Almagest is the Arabized form, with presenter or article "al-", of the Greek superlative μεγίστη: al-Majisti (المجسطي). Gerard of Cremona translated this book from Arabic into Latin in the famous school of Toledo in 1175, which won a great importance in European scientific world.

Moreover, the division of hours into sixty minutes and minutes into sixty seconds only became widespread in use much later, when the man was able to build mechanical clocks to mark that duration, ie, from the sixteenth century , XVII …

It is still an interesting curiosity how in the necessary division of the seconds we do not longer use the sexagesimal system but the decimal : we talk with tenths and hundredths and thousandths of a second

Serve this current coexistence of numbering systems as evidence of possible coexistence also of the dozenal (duodecimal) system, the decimal and sexagesimal in ancient Sumer.

I'll talk another time about the division of time in hours by Greeks and Romans . It is enough now to know that in Greek mythology, Ὣραι (Horai) were originally goddesses who marked the passing of the seasons and they have evolved to be twelve, each one for the twelve divisions of the day.

Why the hour’s dividers are called minutes and...
03 XI 2023.
 
Old November 3rd, 2023 #18
jagd messer
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Cool Why is a circle divided into 360 degrees?

Why is a circle divided into 360 degrees?



The Babylonians 3000 years ago used the sexagesimal system of numbering, which was based upon a multiplication of 6, instead of the decimal system which we now use. They divided the circle for example into 60x6 parts - the 360 degrees. Each degree in turn was divided into 60 parts was again divided into 60.

Claudius Ptolemy took this method of division from the Babylonians and called the first division of the degree the parts minutiae, or a small part. The division of this first small part he called the ‘pars minutiae secundae’ or the second small part, Ptolemy’s names became known as minutes and seconds.

Extracted from the book by title ‘Tick Tock’ compiled by I. Step nova, Prosveshcheniye Publishers, Moscow 1981, Page 36.The Babylonians calculated the number of days in a year as 360.

The earth moved through the Zodiac in 360days as per their estimate and hence the division of the circle into 360 degrees, that is one degree per day. Time measurement is also based upon the sexagesimal system as there are 60 seconds to a minute, sixty minutes to an hour, 24 hours to a day, 30 days to a month and 12 months to an year and also 12 Zodiacal signs in astronomy.

The frequently used angular measurements in Geometry – 30’, 60’, 90’, 180’, & 360' are all multiples of six. The Gradian measure of dividing the quarter of a circle into 100 parts has not become popular. The angular and time measurements remain sexagesimal to this day even though the other two viz. Length and mass of the fundamental measurements have been metricated long back.

Why is a circle divided into 360 degrees?
03 XI 2023.
 
Old November 4th, 2023 #19
jagd messer
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360 is an incredibly abundant number, which means that there are many factors. So it makes it easy to divide the circle into: 2, 3, 4, 5, 6, 8, 9, 10, 12,equal parts. By contrast, 400 gradians cannot even be divided into 3 equal whole-number parts. While this may not necessarily be why 360 was chosen in the first place, it could be one of the reasons we've stuck with the convention.


Why is a full turn of the circle 360°? Why not any other number?

math.stackexchange.com›why-is-a-full-turn-of-the-circle-360-why-not-any-other . .



METRIC SYSTEM SI


Following the French Revolution in 1789, the new government of France, which included the young general, Napoleon Bonaparte, tasked the French Academy of Sciences with replacing the numerous, confusing units of weights and measure with a logical system using multiples of 10.

RELATED: WHY THE IMPERIAL SYSTEM OF MEASUREMENT IS THE WORST

It was decided that the new system should be based on something that was immutable, and the Academy settled on the length of a ten millionth 1/10,000,000 of a quadrant of a great circle of the Earth, as measured around the poles of the meridian that passed through Paris and called a metre.

After six years, a value equal to today's 39.37008 inches was determined, and it was to be called the metre, from the Greek metron, meaning "measure."

By 1795, all metric units were derived from the metre, including the gram for weight, which was equal to one cubic centimetre of water at its maximum density, and the litre which was equal to 1/1,000 of a cubic metre.

Greek prefixes were used for multiples of 10, myria for 10,000, kilo for 1,000, hecto for 100, and deca for 10. Latin prefixes were used for the submultiples, milli for 0.001, centi for 0.01, and deci for 0.1.

Thus, a kilogram equals 1,000 grams, and a millimetre equals 1/1,000 of a metre. In 1799 the metre and the kilogram were cast in platinum, and they remained the standard of measure the for next 90 years. The motto of the metric system was declared to be "for all people, for all time."


The Science of Napoleon - The Metric System and...











 
Old November 6th, 2023 #20
Noel W. Frost
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Working out the Fibonacci series would be a great way to keep the mind occupied in solitary confinement.
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