# fibonacci series

The Fibonacci sequence can help you quickly convert between miles and kilometers

The Fibonacci sequence is a series of numbers where every new number is the sum of the two previous ones in the series.

1, 1, 2, 3, 5, 8, 13, 21, etc.
The next number would be 13 + 21 = 34.

Here’s the thing: 5 mi = 8 km. 8 mi = 13 km. 13 mi = 21 km, and so on.

Edit: You can also do this with multiples of these numbers (e.g. 5*10 = 8*10, 50 mi = 80 km). If you’ve got an odd number that doesn’t fit in the sequence, you can also just round to the nearest Fibonacci number and compensate for this in the answer. E.g. 70 mi ≈ 80 mi. 80 mi = 130 km. Subtract a small value like 15 km to compensate for the rounding, and the end result is 115 km.

This works because the Fibonacci sequence increases following the golden ratio (1:1.618). The ratio between miles and km is 1:1.609, or very, very close to the golden ratio. Hence, the Fibonacci sequence provides very good approximations when converting between km and miles.

New Smash Fighters Announced

Shortly after the announcement of Smash Bros. for Nintendo Switch came the announcement of the new batch of fighters. These are to include:

• Waluigi (Mario Series)
• Air Man (Mega Man Series)
• Prince Sidon (Zelda Series)
• Prince of All Cosmos (Katamari Damacy)
• Master Chief (Halo Series)
• Pyramid Head (Silent Hill Series)
• Primarch Sanguinius (Warhammer 40K Series)
• Jack Sparrow (Pirates of the Caribbean Series)
• Vibeke Dyrsdatter (Valhalla Series)
• Ted Williams (MLB World Series)
• The Unforgiven (Metallica Series)
• 2 Bedroom 3 Bath Kentucky Townhome (ReMax Deluxe Series)
• Dopamine β-hydroxylase (Neurotransmitter Enzymatic Series)
• 144 (Fibonacci Series)

Ridley from the Metroid series is also confirmed not to appear as a playable character.

The Fibonacci Series

In life, the golden ratio is applied almost everywhere! It can be found in anatomy, nature, art, mathematics and music. Consider your body’s ratio the first example… your arms and legs are proportioned to function perfectly with your fingers and toes. Consider nature another example, a tree trunk is big and strong enough to support the branches and leaves. It is all about the “golden proportion” which is “universally regarded as being quite special” because “the proportion that these ratios represent is considered by many to have a certain eye-appeal, balance and beauty” (The Curiosities).

The Fibonacci Series has a basic function of math that allows it to be a structural pattern. As you can tell from the pattern below, each term is the sum of the two preceding terms. This term later became known as a two-term occurrence (Kalman Mena 168).

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377

The history of the Fibonacci sequence goes all the way back to the twelfth century, where Italian Leonardo of Pisa, known as Fibonacci, wrote a book called the Liber Abacci in 1202.  Liber Abacci (Book of Calculation) answered questions posed by mathematicians in Europe.  He related the two-term occurrence of the Fibonacci series 12`to a story about bunny rabbits.  “A closed population begins with one newborn pair of rabbits.  Every month a pair of rabbits produces another pair and the rabbits begin to bear young two months after their birth” (Heyde 1079).  After such research, the application to music uses a combination of analyzing the piece of music and using the Fibonacci formula to determine where the location of the golden section occurs.

The Fibonacci formula consists of the beats in the entire piece multiplied by the Fibonacci Ratio, 0.618.  After calculations, that brings you to the exact beat in the golden section.  Then, to determine a measure number you divide the beat number by beats per measure and that leads you to the measure of golden section. This formula brings the analyzer to the most climatic point in the piece of music.  The most climatic point in the pieces I analyzed were found through the Fibonacci formula.  The measure number, through calculations, was indeed in the Fibonacci series which makes the claim of the golden section valid.

Frédric Chopin’s piece Nocturne in E Flat is a great example to demonstrate the Fibonacci Series function in music. To find the golden section in this piece we must follow the Fibonacci formula.  First, there are 408 beats in the piece and then you multiply 408 beats by the Fibonacci ratio 0.618.  After calculating, the number represents where the Golden section comes in, which is at beat 252. That may seem overwhelming to some to count each and every beat, so I divide 252 by 12, which is the beats per measure and that brings us to measure 21.

As stated before, the number 21 is indeed a Fibonacci number. Nocturne in E Flat most definitely contains the golden section where there is a final return of the theme within the final statement.  After finding the golden section, one can easily find the return of the theme in other measures.  Chopin was strategic in placing the return of the theme because it returns in measures 5, 13, 21 and 34.  Nocturne in E Flat contains the Fibonacci series which allows the piece to be analyzed and calculated by musicians and mathematicians.

Another piece that attracted much attention to its symmetry and poise was Béla Bartók’s masterpiece Musik für Saiteninstrumente, Schlagzeug und Celesta, which translates to Music for String Instruments, Percussion and Celeste.  Along with symmetry in Bartók’s Hungarian Folk material, he applied the Fibonacci numbers to his chord building.  In Music for String Instruments, Percussion and Celeste, the tempo and the meter are changing throughout the entire piece.  The way to calculate this piece is to first, find out how many measures are in the first movement.  In the first movement, there are 88 measures. Then you multiply the 88 measures by the golden ratio .618, which brings you to measure 54.

After analyzing measure 54, there is not enough climatic proof because when looking for the most climatic point in the piece, one would look for any bold gestures such as fortissimo, the highest or lowest range in the entire piece, the farthest removed from the tonic, a cadential gesture or a return of the theme.  Looking closely before and after measure 54, there is climatic proof in measure 55 going into 56.  What signals the analyzer would be the triple fortissimo during the inverted subject.  While listening to Music for String Instruments, Percussion and Celeste, the golden section is aurally prominent for the listener to observe and most importantly measure 55 is a Fibonacci number.

Going back to Frédric Chopin, his piece Prélude Op.28, No.1 was a great example of the golden section taking place on a Fibonacci number.  In Prélude Op.28, No.1, there are a total of 34 measures.  To find the golden section we must multiply the 34 measures by the golden ratio 6.18, which brings you to measure 21.  Measure 21, according to the Fibonacci series, is a Fibonacci number.  To reassure that this is indeed the golden section, one must look for the climatic proof in the piece.  As seen in measure 21, there is a double fortissimo in the bass cleft which supports the golden section theory because there is climatic proof and the number 21 is indeed a Fibonacci number.

From the examples above, it was shown that the Fibonacci series can be applied to music to find the golden section of the piece of music.  By applying this formula, one is able to determine the most climactic point in the piece of music, which is characterized by the bold gestures.  Among many other musicians, Béla Bartók and Frédric Chopin exemplified the Fibonacci series through their well thought out application in their music.  The golden proportion in the pieces analyzed consisted of the eye-appeal, balance and beauty that one would look for in a piece of music.  The golden ratio can be found anywhere, if you look hard enough!

Sources

1. The World Book of Math Power, Volume I, 1993 World Book Encyclopedia, World Book Inc., Chicago, Illinois.
2. Mathematics, Irving Adler, 1990, Doubleday, New York, New York.
3. The Story of a Number, Eli Maor, 1994, Princeton University Press, Chichester, West Sussex.
4. The Fibonacci Numbers: Exposed, Dan Kalman and Robert Mena, Mathematics Magazine, Vol. 76, No. 3 (Jun., 2003), pp. 167-181
5. Functions, Scales, Abstract Systems and Contextual Hierarchies in the Music of Bartók, The Music of Béla Bartók by Paul Wilson, Review by: Michael Russ, Music & Letters , Vol. 75, No. 3 (Aug., 1994), pp. 401-425
6. On a Probabilistic Analogue of the Fibonacci Sequence, C. C. Heyde, Journal of Applied Probability , Vol. 17, No. 4 (Dec., 1980), pp. 1079-1082
7. Fibonacci’s Forgotten Number, Ezra Brown and Jason C. Brunson, The College Mathematics Journal , Vol. 39, No. 2 (Mar., 2008), pp. 112-120

2

Jack Storms makes beautiful cold glass geometric sculptures. One of the few people who has this ability to grind out these beautiful geometric pieces (based in the Fibonacci series). Each piece takes about 8 to 18 weeks to produce by cutting and splicing dichroic glass along with grinding.

The Fibonacci sequence can help you quickly convert between miles and kilometers

The Fibonacci sequence is a series of numbers where every new number is the sum of the two previous ones in the series.

1, 1, 2, 3, 5, 8, 13, 21, etc.
The next number would be 13 + 21 = 34.

Here’s the thing: 5 mi = 8 km. 8 mi = 13 km. 13 mi = 21 km, and so on.

Edit: You can also do this with multiples of these numbers (e.g. 5*10 = 8*10, 50 mi = 80 km). If you’ve got an odd number that doesn’t fit in the sequence, you can also just round to the nearest Fibonacci number and compensate for this in the answer. E.g. 70 mi ≈ 80 mi. 80 mi = 130 km. Subtract a small value like 15 km to compensate for the rounding, and the end result is 115 km.

This works because the Fibonacci sequence increases following the golden ratio (1:1.618). The ratio between miles and km is 1:1.609, or very, very close to the golden ratio. Hence, the Fibonacci sequence provides very good approximations when converting between km and miles.

# Fibonacci sequence in music

Submitted by Ridingunicorn:

This is an art project I did a while back. The entire thing was based on the Fibonacci series. The spiral is a golden spiral, and the mark itself was numbers of the series.  This portrait took me about 2 months time with a break of 2 weeks in the middle.

The Fibonacci sequence can help you quickly convert between miles and kilometers

The Fibonacci sequence is a series of numbers where every new number is the sum of the two previous ones in the series.

1, 1, 2, 3, 5, 8, 13, 21, etc.
The next number would be 13 + 21 = 34.

Here’s the thing: 5 mi = 8 km. 8 mi = 13 km. 13 mi = 21 km, and so on.

Edit: You can also do this with multiples of these numbers (e.g. 5*10 = 8*10, 50 mi = 80 km). If you’ve got an odd number that doesn’t fit in the sequence, you can also just round to the nearest Fibonacci number and compensate for this in the answer. E.g. 70 mi ≈ 80 mi. 80 mi = 130 km. Subtract a small value like 15 km to compensate for the rounding, and the end result is 115 km.

This works because the Fibonacci sequence increases following the golden ratio (1:1.618). The ratio between miles and km is 1:1.609, or very, very close to the golden ratio. Hence, the Fibonacci sequence provides very good approximations when converting between km and miles.

Fi"bunn"acci Sequence - Animated gif version

La ira de fibonacci

En una serie de secuencias, rayando lo complejo, tu cuerpo juega en lo absurdo emitiéndose en mi lo irracional de tus reflexiones. Tu figura que escapa de lo convencional al vulgo antinatura, y más que besos lo tuyo son semillas de donde brota lo aureo, lo aureo de tu belleza.

Fibonacci Cubs

Fibonacci sequence in real life. Happy #FibonacciDay! 0⃣1⃣1⃣2⃣3⃣5⃣8⃣… 🌀infinite.

#nov23rd #fibonacci #series #sequence #spiral #math #numbers #lovelovelove #lovemath #welovenumbers #mathematics #formula #integers #recurrence #reallife #examples #marketing #branding #strategy #digital #online #socialmedia #smm #design #technology #analytics #storytelling #advertising #TheDigitalSavvy https://www.instagram.com/p/Bb1yVMBh9AH/

4

The relative frequency of each number in the first 1,000 digits of:

1. pi (the ratio of a circle’s circumference to it’s diameter)
2. e or Euler’s number (the limit of (1 + 1/n)n  as n approaches infinity)
3. √2 or Pythagoras’ constant (the length of the hypotenuse of a right triangle with legs of length 1)
4. phi or the golden ratio (One number in the Fibonacci series divided by the number that precedes it in the series.)

Mary Martin, Climbing Form, 1954-62

Fibonacci series

The greatest pattern there is. Why?

You can find fibonacci numbers anywhere in nature. Somehow, these numbers just seem occur. Interestingly, there are 8 planets in our solar system.

I love this pattern because it is simple yet astounding nonetheless. When you get really high in the series, the numbers conform to the golden ratio. That is, if you divide a number by its predecessor, the answer you will get is the golden ratio.

The golden ratio is believed to be aesthetically pleasing. For centuries it has been used in architecture and fine art, and it is used to analyse financial markets.

Some everyday occurrences:

• 5 fingers, 5 toes
• 3 leaf clover
• 13 segments in an orange
• Centipedes have 34 legs, spiders have 8

This list could be made very long, but it is quite a monotonous task. I am sure you can claim there are places where fibonacci numbers don’t occur, but take a closer look…

Things that have particular significance to me:

• The neurocranium, the braincase of our skull, is made up of 8 bones. Protection for the brain, ourselves.
• The bee ancestry code. A male bee has 1 parent, 2 grandparents, 3 great grandparents, 5 great great grandparents and so on. I don’t like bees, one sting could kill me, but that bee has a fibonacci lineage, leading to infinity.
• Life incorporates it.
• Every fibonacci number has a prime factor that is not a factor of any other fibonacci number.

In my tattoo, the numbers I have used are: 1,1,2,3,5,8,13,21,34,55,34,21,13,8,5,3,2,1,1. There are gaps of 5 in-between. The total length is 321mm.

• 55 has a prime factor of 11, my birthday. The other is 5, a fibonacci number that is used a lot in the design. 55 is the highest number to be featured, and is depicted as a square.
• A length containing 5 appears 21 times in the tattoo (18 gaps of 5, 2 lines of 5, and a square of 55).
• I am 21 and was 21 when I got the tattoo, on the 13th day of the month of my birthday, the 4th month of 2011 (2011 contains the first 4 fibonacci numbers 0,1,1,2).
• 1012,1021,1102,1201,2011. 2011 is the 5th year ever to contain the first 4 fibonacci numbers.
• (1+1+2+3+5+8+13+21+34+55+34+21+13+8+5+3+2+1+1)/11=21.
• (1+1+2+3+5+8+13+21+34+55)/11=13
• Any 10 consecutive numbers in the fibonacci series divided by 11 give a whole number.
• Randomly, the tattooing took exactly 3 hours.

math:: The Fibonacci Series: When Math Turns Golden

Esther Inglis-Arkell — The Fibonacci Series, a set of numbers that increases rapidly, began as a medieval math joke about how fast rabbits breed. But it’s became a source of insight into art, architecture, nature, and efficiency. This mathematical game explains the structures of leaves and lungs, is replicated in paintings and photographs, and pops up as the basis for the pyramids, the Parthenon, and packing efficiency. Find out where the Fibonacci Sequence comes from and why it keeps eerily showing up.

The Origin of the Series:

The Fibonacci Series gets its name from Leonardo Fibonacci, who lived in the twelfth century. He wanted to calculate the ideal expansion of pairs of rabbits over a year. He assumed that each pair would produce another pair as soon as they matured at one month. In January, a new pair of rabbits would be born (1) they would reach maturity in a February (1) and breed, producing a new pair in March (2). They would then breed again, and produce a new pair in April (3), and another pair in May. Meanwhile, they rabbits born in March would reach maturity in April so in May would see two new pairs of bunnies produced, bringing it to a total of 5 pairs. Now the rabbits born in January, March, and April would all be adding new pairs, bringing June’s total to 8 pairs..

The expansion would carry forward, with each new pair coming to maturity and starting their own little Fibonacci Series to be added to the whole. Over the months, with no deaths, the rabbit pair expansion would look like this:

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

Anyone can see that by December the poor owner would be inundated with rabbits. Sharp-eyed readers can also see that each new number in the sequence is the combination of the two numbers before it. Five plus eight makes thirteen. Eight plus thirteen makes twenty-one, and so on.

Fibonacci Goes Gold in Art and Architecture:

Many would respond to this with a shrug and a mental note to not let Fibonacci near any of their rabbits. It turns out, though, that he was really on to something. Mathematicians and artists took this sequence of number and coated it in gold. The first step was taking each number in the series and dividing it by the previous number. At first the results don’t look special. One divided by one is one. Two divided by one is two. Three divided by two is 1.5. Riveting stuff. But as the sequence increases something strange begins to happen. Five divided by three is 1.666. Eight divided by five is 1.6. Thirteen divided by eight is 1.625. Twenty-one divided by thirteen is 1.615.

As the series goes on, the ratio of the latest number to the last number zeroes in on 1.618. It approaches 1.618, getting increasingly accurate, but never quite reaching that ratio. This was called The Golden Mean, or The Divine Proportion, and it seems to be everywhere in art and architecture.

The Greeks used the 1.618 proportion to construct The Golden Rectangle. It was a rectangle with sides measuring one and 1.618 (or with side measuring to consecutive Fibonacci Numbers). This was considered the most mathematically beautiful structure, and frequently used in architecture. The Parthenon incorporates a number of Golden Rectangles into its structure and decoration. What’s more, the Pyramids have their own Divine Proportions. If the base of the Pyramids is considered one unit, the sloping sides are 1.618 units, and the height is the square root of 1.618 units high.

Today, many photographs and paintings use golden proportions. Take a Golden Rectangle, or a rectangle in which the two sides are Fibonacci Numbers. That rectangle can be chopped up into smaller rectangles and squares that all also have Fibonacci proportions. Many works of art contain objects that fit within these proportions.

But what about curves? That’s where the Fibonacci sequence really shines. Draw an arc from one corner of those nested squares to the opposite corner. Do it enough, with increasingly nested squares, and it makes a Golden Spiral. The spiral is used in art, but it’s seen even more often outside of a gallery.

Fibonacci Spirals in Nature:

So far, the Fibonacci Series has been popping up solely as a result of humans going crazy for a certain series of numbers. Although the original problem was illustrated with rabbits, anyone knows that population doesn’t actually expand that way. Rabbits don’t always get born in pairs, and even though they’re famous for their fertility, they don’t conceive every time they try.

In fact, the best examples of real world Fibonacci Series are found in the plant kingdom. Many plants that branch outwards towards the sun do so in branches equal to Fibonacci numbers. The original sprig comes up from the earth. For the first period of time, it just sprouts upwards. Then it develops meristem points - points from which new branches can form - and those sprout into two separate branches. Those branches push upwards for another period of time, and then develop two points of their own. The overall number of sprouting points develops in a Fibonacci Series.

The most celebrated example of the Fibonacci Series is the spirals it creates. Florets on a cauliflowers, fruitlets on a pineapple, seeds on a sunflower, they all spiral outwards. And each of those spirals contains a number of seeds, florets, bumps, leaves or tubercles that are equal to a Fibonacci number. Some might say that humans pick and choose, deciding to ignore the flowers that do not spin out their seeds in a Fibonacci series, but it turns out there’s a reason for the repeated Fibonacci numbers on different species of spiraling plant - it’s the perfect way to pack.

As a sunflower bulb develops seeds, it has to give each of its potential offspring equal space to do flourish. They need to be packed as evenly and equally as possible. But spirals aren’t the best way to fit seeds into a space, so why do plants do it? Because they don’t build a space and then pack it full of seeds like humans do a warehouse. They make seeds as the bulb that those seeds mature in expands.

Plants make these spiralling seedpods by maturing seeds at the center and the stretching the space the seeds inhabit outwards. If they deposited the seeds one directly beneath another, the seeds would be squished on top and bottom and have space to the sides - the flower pod would become an elongated seed holder and would strain its stem. And so the flower matures seeds in a circular pattern, setting the growing seeds at an angle to each other and letting them expand outwards. But what’s the most efficient way to do that? If the flower made four seeds for every completely circular ‘turn’, the fourth seed would be deposited right underneath the first seed - making four rows of seeds that push outwards. That’s better than a line, but still not an efficient use of space. It turns out that the best number of seeds to deposit per turn is around 1.618. Since flowers can only make whole numbers of seeds, this means that no seed is lodged directly under its predecessor. Instead, the seeds spiral outwards from the center.

Since these spirals have the Divine Proportion of 1.618 seeds per turn, counting the seeds any spiral will result in getting a Fibonacci Number. The Fibonacci Series is also seen in plants that put out leaves from one stalk. As the plant grows upwards, the number of leaves per completed by the end of each 'turn’ around the stem will be a Fibonacci number.

Fibonacci For Fun:

When math people get ahold of something they never let go. The Fibonacci Series has a whole lot of strange, and interesting, quirks. For example, the sum of any 10 consecutive numbers in the Fibonacci Series is divisible by 11. The square of a Fibonacci Number, minus the square the Fibonacci Number two terms before it, will yield another Fibonacci Number. Piano keys in an octave are made up of Fibonaccie Numbers; eight white, five black, and thirteen in all.

If you are subscription fee curious about the many Fibonaccis silently lurking around the world, The Fibonacci Quarterly is a paper that is devoted entirely to the Fibonacci Series. Don’t worry about them running out of material. The editor says they have a backlog of waiting articles. The series is everywhere.

Via Google, Scientific American Book Club, Time, Math Is Fun, World Mysteries, Maths Surrey,Again, and Again, and Again.