Fingers

Fingers were man's earliest calculating device. By increasing or decreasing the number of fingers he displayed he was able to perform simple subtraction and addition. But finger counting has its limitations so man developed more sophisticated devices for counting.

Quipu

The Quipu was the very first analog computer used by Incan tribes (ancient Indian civilization in North America). The computation was done by counting the number of knots.

Abacus

The Abacus was the next analog computer which was developed in China in the 12th Century A.D. The device has a frame with its beads strung on wires or rods. Arithmetic calculations are performed by manipulating the beads. It was also known as Soroban in Japan and Suan Pan in China. To these days, old folks still use this device despite the presence of modern calculator especially at China Towns in different countries. The old folks are used to it compare to calculators.

Napier's Bones

Napier's Bones is composed of eleven rods with numbers marked on them in such a way that by simply placing the rods side by side products and quotients - large numbers can be obtained. The sticks were called bones because it is made of bone or ivory. It was named after Scottish mathematician John Napier, who is famous for his invention of logarithms.

(Picture from http://cc.wsd3.org/media/images/piano_keys.jpg)

            When the word music comes to mind, many people think of their favorite hip-hop artists, rappers, and guitarists.  Many people have at one point or another played an instrument in a band, learned piano, or have sang.  Notes elegantly drawn across the page can calm and sooth people, or get them in the zone for sporting events.  Music brings joy to those who play, write, or listen.  Ask them if it is connected to mathematics, and many people will be puzzled.  However, once the shock recedes and they think about it, they realize that there are striking similarities.  Not only the obvious ones, like beats in a measure, but the wavelengths and ratios between notes.  Studies have shown that babies who listen the classical music can grasp mathematical concepts quicker than those who did not.  In essence, math is creative and beautiful, which is why such elegant connections are made between them.  This paper will attempt to discuss the simple and complex connections between the two studies as well as attempt to uncover new ideas in the fields of music, such as a wind instrument with a piano’s range. (Background from http://www.math.niu.edu/~rusin/uses-math/music/ and http://www.woodpecker.com/writing/essays/math+music.html)

Music is made up of beats.  Beats are pulses in which time is marked.  The most common measure has four beats in it, which means no matter the combination of notes, they must add up to four beats.  For example

(image from http://www.uoregon.edu/~kford/picturegallery/intromusic_files/image005.gif)

ü      Level one is called a whole note, meaning that this note will get four beats.  1note @ 4 beats

ü      Level two is composed of two half notes, which are two beats each. 1 note @ 2 beats

ü      Level three is 4 quarter notes representing one beat each.  1 note @ 1 beat.

ü      Level four is 8 8^th notes, which equal one half a beat each.  1 note @ .5 beats

ü      Level five shows 16 16^th notes, which would each receive one quarter of a beat.  1 note @ .25 beats.

The same holds true for rests: one whole rest equals two half rests equals four quarter rests, and so on and so forth.  The following picture shows the relationships between rests and beats.

(picture from http://www.enchantedlearning.com/music/label/notesrests/answers.GIF)

(This and similar pictures from http://www.dkimages.com/discover/previews/756/223171.JPG)

            Basics first.  Before attempting to play music, any musician will look at the time signature.  The top number tells a musician how many beats are in one measure and the bottom one reveals which note will get one beat.  In the above example, there will be four beats in one measure, and a quarter note (1/4 a measure) will get one beat.  But this does not mean that only quarter notes can take be in that measure.  Take a look at the first measure above

(From http://www.dkimages.com/discover/previews/756/223171.JPG, edited by myself)

Another simple aspect of music is the tempo.  Tempo is recorded in beats per minute and tells the conductor and musicians how fast a piece of music should go.  This is marked at the beginning of a piece above the staff (see above).  A tempo of 60 is very slow at one beat per second while a tempo of 180 is very fast at three beats per second.  The formula to figure out how long a song will be is:

60S=MB

        T

S equals time in seconds, m equals the number of measures, b equals the beats in one and t equals the tempo

            The key signature tells the musician which keys to play in.  Different keys are made up of different notes even though the ratios to the first note remain the same.

(image from http://www.ericweisstein.com/encyclopedias/music/wimg3.gif)

Take any note, and the transition to the note directly to the left or right of it is called a half step.  The note two notes to the left or right is called a whole step.  For example, take “F”.  An “F#” and an “E” are considered half steps, while “G” and “D#” are considered whole steps.  To create a major key we used this pattern.  Start with any note and then:

+1 +1 +½ +1 +1 +1 +½

which will give you the key of that note.  This will also give you a major scale.    According to www.answer.com, majors scales are defined as: “an ascending or descending collection of pitches proceeding by a specified scheme of intervals”.  This interval is:  x+1+1+½+1+1+1+½, with x representing any note, which was discussed earlier.  There are twelve major scales that are distinct, C, C#/Db, D, Eb, E, F, F#/Gb, G, Ab, A, Bb, B/Cb.  ( # represent sharp, b represents flat).  Any scales after these would be defined as octaves, one whole pattern above the others.  To find octaves we use the formula (with x representing the first note of any scale):

x=x+11

Take the formula for the major scale again:

x +1 +1 +½ +1 +1 +1 +½

Apply this to any note and every individual answer you get along the way makes up the key.  For example, start on Bb.

            For a major scale:                       Bb +1 +1 +½ +1 +1 +1+½

            Which corresponds to:                Bb,C, D, Eb,  F, G, A,  Bb  

            Which gives us the key of Bb:     Bb, C, D, Eb, F, G, A, Bb

There is also something called a minor scale, which involves lowering certain degrees in the major scale.  This makes the scale sound sad, and is characteristic of many slow, sad songs as well as ominous sounding classical music.  To get a minor scale, we lower the third, sixth and seventh note of the major scale by a half step

            Where as the scale used to be:        x  +1  +1  +½  +1  +1  +1 +½

                                                                  1   2    3    4      5    6    7    8

            The minor scale becomes:                x  +1 +½  +½  +1 +½ +½ +½

            So in the major scale of C:             C, D,  E,  F, G,  A,  B,   C

                                                                 1  2    3    4   5   6    7    8

            We get the C minor scale:              C, D, Eb, F, G, Ab, Bb, C

            Now take the Eb major scale          Eb +1 +1 +½ +1 +1 +1+½

            In order to get:                                Eb, F, G, Ab, Bb,C, D, Eb

            Notice that the scales of Eb major and C minor have the same notes, but in a different order:

                                                            C,  D, Eb, F,  G, Ab, Bb, C

                                                            Eb, F, G, Ab, Bb, C,  D, Eb

            This is because C major and Eb minor are what is called relative keys.  Eb is the relative minor to the key of C.  This may seem confusing, but it is easily summarized with what musicians call the “Circle of Fifths” (shown below).  Although they keys on a piano (the white ones) read; C, D, E, F, G, A, B; musicians memorize the notes according to their structure in the circle of fifths: F, C, G, D, A, E, B.  The scales are organized this way because this is the way the number of sharps and flats that appear in the key are organized.  Forget about the F for a moment:

Circle of Fifths:                             F, C, G, D, A, E, B

Scale(key) number                            1,  2,  3,  4, 5, 6

                                                    7

Number of Sharps:                            0,  1,  2,  3, 4, 5

                                                    6,  7

The first scale (key) we start with is C.  C has no sharps (0).  Next we go to G.  G has one sharp (1).  D has 2 and E has 3.  In order to find out what these sharps are, we start with F and count the number.  So G, with one sharp, gives us only F.  This means that F is the only note that is sharp in the key of G.  Take the key of E.  E has 4 sharps.  Starting with F, we get F, C, G, and D.  This means that the four sharps in the key of E are F, C, G, and D.

            However, this gets complicated when the circle goes on past B after reading clockwise.  The notes F# and C# appear.  These notes were not in the original sequence of letters. This is because the that sequence is read both forwards and backwards (which we’ll discuss later) and that F# and C# do not fit when it is read backwards.  To help the reader understand this, imagine there are two separate sequences, one for the sharp notes, and one for the flat notes.  So in order to include F# and C#, the sharp sequence should look like this:

                        F, C, G, D, A, E, B, F#, C#.

            So therefore, following the same method discussed above, A C# scale would have seven sharps (remember to start counting at C and not at F).  These seven sharps are F, C, G, D, A, E, and B.  Which gives us the scale and key of C sharp major: C#, D#, E# (played as F), F#, G#, A#, B# (played as C).

            Now the circle of fifths also works for the flat scales.  Just drop the F# and C#, and reverse it and add an F to the beginning:

            F, B, E, A, D, G, C, F

            1,  2, 3,  4, 5,  6,  7

            This is the “Circle of Fifth” for the flat scales.  F is an anomaly here because it is the only major scale to use flat notes without itself being flat.  Take a look:

            F is the first flat scale.  Because F is the first flat scale, it gets one flat.  To figure out what particular note is flat, start counting from the beginning, always skipping the first F.  Therefore the one and only flat in the key of F major is a Bb.  Let see:

                        F  +1  +1  +½  +1  +1  +1 +½

                        F  G    A   Bb   C    D   E    F

            Here it is easy to see that the B note is indeed the only flat note in the F major scale.  As another example, the key of Gb.  G is the sixth note in the circle and therefore gets six flats.  Starting from the beginning and leaving off the first F, we get that the six flats are Bb, Eb, Ab, Db, Gb, and Cb, meaning that the B, E, A, D, G, and C all become flat in the key of Gb:     

Gb, Ab, Bb, Cb (played as B) Db, Eb, F, Gb

            The inner circle of the “Circle of Fifths” also described what the “relative minor” is.  “C” on the inner circle corresponds to “Eb” on the outer circle.  This means that C minor uses the same notes as they key of Eb, except the scale starts with C and not with Eb, as discussed earlier.  This holds true for all notes.  Therefore, “A minor” uses the same notes as “C” and so on and so forth.  Just as patterns are found with numbers, so too can they be found in music.  The circle of fifths is a table that summarizes the answers to the major and minor formula.

Now this pattern is the same for all notes and scales.  To transpose a scale or change it to a different key, you use a method similar to a shift cipher.  For instance; take the notes C, D, E, F, and G in the key of C (no sharps of flats).  If one wants to transpose these notes, one first attaches numerical values to the notes based on their scale.  C is the first note of the scale so it gets a value of one.  D is the third note so it gets a value of 3:

   C   D   E   F   G

      1    3    5   6    8  

Then one takes a new key, say G(one sharp only which is an F).  Due to the fact that G is first note in a the G scale, it equals the note value of one given to C in the previous example.  A is the third note in the G scale which matches up with the third note in the C scale: D.

   C   D   E   F   G

      1    3    5   6    8  

         G   A   B   C   D     

The transposition is complete.  The spaces between CDEFG will be the same as GABCD.  This is exactly the same as transformations on a graph.  Under a transformation, the image looks the same, with the same distance between points.  This is the same as transposing; the piece sounds the same, just with different points.

Above, whole steps, half steps and keys were discussed.  The reason artists use keys is so that notes in piece sound pleasant to a listener.  It may seem difficult to makes people’s taste connect to math.  However, there is a way to determine what will make notes sound sweet to one’s ear.  It all starts with a Hertz (Hz).

            A Hertz is a measure of frequency.  Hertz is a ratio to radians per second.  The more radians per second in a sound wave, the higher the pitch or frequency.  One radian per second is equal to:

1radian/second= 1  Hz

      2

(http://en.wikipedia.org/wiki/Radian_per_second)

            Pythagoras, one of the most influential mathematicians of all time, was one of the first mathematicians to make connections between music and math.  He believed that everything in life could be expressed as a fraction.  He took this approach towards music as well.  Pythagoras lived during Ancient Greek times.  During this era, music was much simpler than it is today.  The octave had only five notes.  Pythagoras noted that each note was a fraction of a string.  Take for example the double bass, a string instrument.  The last string on a double bass, when bowed or plucked, vibrates at 196 Hz (which corresponds to the note “G”).  If one places their finger about ¾ of the way down the string and presses down, the note becomes a C, which is approximately 4/3 (the reciprocal) of the frequency, at 262 Hz. 

            This “reciprocal rule of music” applies to all notes of all non-electric instruments.  When the no valves of a trumpet are pressed, a “C” is heard, (corresponding to C₄ on the chart below).  When the first valve is pressed, a “G” is heard, (corresponding to G₄ on the chart below).  “C” has a frequency of 261.63 and “G” has a frequency of 393.

    393__   _{is approximately equal to: } _3_

 261.63                                          2

Being that trumpets make sound by air vibrations, trumpets change notes by changing the amount of space the air has to vibrate.  By pressing down the first valve, the air space is cut by 2/3, the reciprocal.  The ratios between the frequencies of notes always equal the reciprocal of the fraction of space the note can vibrate in.  The ratio between C and G is always 3 to 2, and therefore to play a G, one must take 2/3 of the space used to create the sound, whether it be 2/3 of the string, 2/3 of the air tube, 2/3 the area of the block (on a xylophone).

 Note 	Frequency (Hz)	 Note 	Frequency (Hz)	 Note 	Frequency (Hz)	 Note 	Frequency (Hz)
C0	16.35	C2	65.41	C4	261.63	C6	1046.5
C#0/Db0	17.32	C#2/Db2	69.3	C#4/Db4	277.18	C#6/Db6	1108.73
D0	18.35	D2	73.42	D4	293.66	D6	1174.66
D#0/Eb0	19.45	D#2/Eb2	77.78	D#4/Eb4	311.13	D#6/Eb6	1244.51
E0	20.6	E2	82.41	E4	329.63	E6	1318.51
F0	21.83	F2	87.31	F4	349.23	F6	1396.91
F#0/Gb0	23.12	F#2/Gb2	92.5	F#4/Gb4	369.99	F#6/Gb6	1479.98
G0	24.5	G2	98	G4	392	G6	1567.98
G#0/Ab0	25.96	G#2/Ab2	103.83	G#4/Ab4	415.3	G#6/Ab6	1661.22
A0	27.5	A2	110	A4	440	A6	1760
A#0/Bb0	29.14	A#2/Bb2	116.54	A#4/Bb4	466.16	A#6/Bb6	1864.66
B0	30.87	B2	123.47	B4	493.88	B6	1975.53
C1	32.7	C3	130.81	C5	523.25	C7	2093
C#1/Db1	34.65	C#3/Db3	138.59	C#5/Db5	554.37	C#7/Db7	2217.46
D1	36.71	D3	146.83	D5	587.33	D7	2349.32
D#1/Eb1	38.89	D#3/Eb3	155.56	D#5/Eb5	622.25	D#7/Eb7	2489.02
E1	41.2	E3	164.81	E5	659.26	E7	2637.02
F1	43.65	F3	174.61	F5	698.46	F7	2793.83
F#1/Gb1	46.25	F#3/Gb3	185	F#5/Gb5	739.99	F#7/Gb7	2959.96
G1	49	G3	196	G5	783.99	G7	3135.96
G#1/Ab1	51.91	G#3/Ab3	207.65	G#5/Ab5	830.61	G#7/Ab7	3322.44
A1	55	A3	220	A5	880	A7	3520
A#1/Bb1	58.27	A#3/Bb3	233.08	A#5/Bb5	932.33	A#7/Bb7	3729.31
B1	61.74	B3	246.94	B5	987.77	B7	3951.07

(A partial listing of all musical note frequencies.  From http://www.phy.mtu.edu/~suits/notefreqs.html, edited by Alex Donnelly)

In music, the lowest note possible to play is a low “C” at 16.35 hertz and G# is the highest note at 13289.75 hertz.  There are two constants involved with the frequency of notes.  The first constant is that the A above middle C (A₄) is equal to 440Hz.  The second constant is the relationship between each consecutive note.  There is a number that each note is multiplied by to get each successive note.  This number is 2 to the power of 1/12 or 1.059463094.  To get each successive note, a formula similar to the population or interest formula is used.

F_new=F_old ∙ (1.059463094)ⁿ

(Where F is equal to the frequency and n is equal to the number of half-steps of half-tones [the space between C and C# for example])

For instance, to find the frequency of the note 12 half-steps above A₄, multiply 440Hz by 1.059463094 to the power of 12.  Because 1.059463094 is equal to 2 to the power of 1/12, this simplifies to be 2, therefore 440Hz times 2, to give an answer of 880Hz.

F= 440{A₄} ∙ (2^1/12)¹²  =  880Hz{A₅}

The difference between A₄ and A₅ is what as known as an octave.  The difference in frequency between this octave is 440Hz and 880Hz.  This octave is 2 times the first note.  The same is true for all octaves, which you can tell from the chart above.  Therefore, all octaves have a ratio between them of 1:2.

Note that these ratios can be applied to all notes.  The difference between two notes is always the same ratio..  For instance, middle C (C₄₎ has a hertz of about 262 and the G above it (G₄) is 392hz.  When you divide the two, you get the ratio between the notes: 3/2 (example number 7 below).  The difference between C_n and G_n will always remain in proportion to eachother in the way of 3:2.  This means that every 3 radians of C will match up with every 2 radians of G at the x-axis.  This is a relatively low ratio, which sounds good to our ears.  The ratio between C and C# (about 262Hz and 277Hz respectively) is 135:128.  This is a huge ratio, which sounds terrible to a listener.  (Research from http://en.wikipedia.org/wiki/Hertz, http://en.wikipedia.org/wiki/Pitch_%28music%29) (Picture below from http://en.wikipedia.org/wiki/Mathematics_and_music)

Note	Ratio	Interval
0	1:1	unison
1	135:128	major chroma or minor second
2	9:8	major second
3	6:5	minor third
4	5:4	major third
5	4:3	perfect fourth
6	45:32	diatonic tritone
7	3:2	perfect fifth
8	8:5	minor sixth
9	27:16	Pythagorean major sixth
10	9:5	minor seventh
11	15:8	major seventh
12	2:1	octave
 

Notice how every third C (green) matches up with every second G (red), making the notes sound nice.  (picture above and below taken from http://www.musicmasterworks.com/WhereMathMeetsMusic.html)

Here, F sharp is red, which matches up with green C.  These notes will only match up every 45^th time for C and 32^nd time for F#, making them sound unpleasant.  The above figures are sine curves, starting at the origin, which are used to graph function is math, but also to map out sound waves in music, one of the deeper connections thus far.  ( Idea from Jessica Seminelli in Eureka!, “The Symphony of Sine Curves”.  Research compiled from http://www.musicmasterworks.com/WhereMathMeetsMusic.html and http://thinkzone.wlonk.com/Music/12Tone.htm)

 Harmonics are also a little on the complicated side.  Harmonics the vibrations and are what makes notes on a flute and a violin sound different.  These two distinct instruments make vibrations in very different methods.  The strings vibrate when bowed in string instruments.  The density, length, and tension of the string impact the vibrations made.  In a flute, the air vibrates when a player blows across a narrow opening, similar to blowing across a water bottle.  Brass instruments make vibrations from one’s lips.  Reeds instruments make vibrations between the reed an the mouth piece, similar to a kazoo.  Finally, harmonicas make sound by the little strips of metal vibrating quickly.  The less area there is where the vibrations are made determine pitch.  A flute, which is a very small instrument, has very little are to vibrate, creating notes with very high pitch.  A tuba a an extremely large are in which the air can vibrate, creating notes with very low pitch.  One can alter the vibrations by elevating of lowering the space the vibrations have to expand.  A violin player moves his or her fingers up or down the strings to make them smaller or larger, affecting pitch.  A brass player presses and releases valves, which allow the air to flow into larger of smaller chambers, affecting pitch.  Reed players open or close holes in their instrument, allowing air to escape earlier or later than normal.  The physical properties of each instrument limit their range, because no instrument out on the market can have accommodations for every note.  Keep in mind that players do not drill openings in their instrument, but rather they are already there, making a definitive range for that instrument.  Therefore, the instrument with the largest range is a piano.  (http://en.wikipedia.org/wiki/Mathematics_and_music)

No wind instrument has ever been created that can mimic the range of the piano. This is limiting because pianos are expensive and are not easily transported.  They also have to have their inner workings inspected after every move, and once a year to insure they are pitch perfect.  I will try to solve this problem by creating a wind instrument that can cover the range of a piano.

To start with,  I had to look at which instrument have the extreme low and high ranges, because there are not many instruments that can do this.  Now there are trombones that exist that have a trigger, allowing them to cover more notes towards the lower reaches of the piano.  Taking form those advancement, a trigger could b e used, located near the mouthpiece, to switch from a higher end of an instrument the lower end.  They also have to be of the same type, say two brass instruments, because they need to share a mouthpiece.

I started with finding instruments with far reaches in terms of pitch.  The following picture is the range of many instruments.  (image below from http://upload.wikimedia.org/wikipedia/en/timeline/d509b142febba1b37c8544b544ff1ac5.png)

The octocontrabass clarinet’s range can not be covered by the chart, as the instrument can reach the A below the last part of the chart.  So if a octocontrabrass clarinet and a soprano clarinet were combines, they would almost cover the entire range of a piano.  Notice how the high reach of the octocontrabass clarinet and the low reach of the soprano clarinet math up, so it would be a seemless connection.  The shortcoming is, the combined range is still 6 notes short of a piano’s range, however it can reach 4 notes lower than a piano can.  I believe that this is close enough.

octocontrabass clarinet (on left) (image from http://www.jayeaston.com/images/octocontra-Clarinets-Leblan.jpg)

Soprano Clarinet ( image from http://www.interstatemusic.com/wcsstore/InterstateMusic/ims/ipf/105773.jpg)

When combined, these two instruments would be connected at the mouth piece, with the smaller of the two sticking out at about a 45 degree angle to the ground.

            In conclusion, there is a very elegant connection between math and music.  Whether is be simple, such as beats in a measure.  Or more complex, sine curves and hertz ratios, it is clear why they are so similar.  The different measure connect to the patterns learned in early math, 1-2-3-4, 1-2-3-4,1-2-3-4.  Just like mathematical sequences have patterns so does music.  The different number of beats a note gets connects to the fractions in math. 1/1 equals 4/4.  Simplification.  No matter what, each measure will always have four beats in it, just like 4/4 and 1/1 are equal, the answer always being one.  The ratios between frequencies, are also similar to word problems in early math where one objects would complete a task every two minutes, and another every three, and a student is asked when they will meet up.  This is an early concept of the least common denominator.  When attempting to add fractions without the same denominator, a student must find the LCD, just as the frequencies, make music more pleasant to people.  The connection between transposing and translating were shown, as if the mold was picked up and moved somewhere else to create almost identical music.  And finally, an pretty successful attempt was made to design a wind instrument that could cover the range of a piano.  All in all, the connections between math and music are deep.  Perhaps that is why musicians are so good at math.

Phi can be expressed using a line segment. We make it such that the ratio of B to A+B is equal to the ratio of A to B.

Many designs - used in buildings, sculptures and paintings use the Golden Ratio for their dimensions. Architects and artists tend to use them often as they are considered very pleasing to the eye.

For example, the Parthenon uses the golden ratio for its construction.

Another example is the Mona Lisa painting.

The golden ratio also determines how attractive a person is.

For example, the American pop singer and actress Jessica Simpson is attractive because the proportion of her face fits geometrically on the human face mask which conforms to some aspect of the Golden Ratio.

Golden Rectangle

This rectangle is a special rectangle where the ratio of the length to the width is the Golden Ratio, which is .

When a square is cut off from the golden rectangle, the new rectangle is still similar to the original rectangle.

Below is an illustration:

and so on…

Golden Triangle

The golden triangle is a special isosceles triangle. The top angle is 36° while the bottom two angles are 72° each. We then bisect one of these base angles. The resulting blue triangle:

is also a golden triangle! Thus, we can keep bisecting the base angle to get a set of whirling triangles:

From this, we can draw a logarithmic spiral similar to the Fibonacci spiral:

Pentagram

A pentagram is a star-shaped figure which is made out of the five diagonals on a pentagon:

As can be seen, a pentagram has five sides. From three of these sides, we can get a couple of different golden triangles:

From the figure below:

we can also conclude that two non-consecutive sides of a pentagram divide each other in the Golden Ratio.
Fibonacci number

Fibonacci number

The Fibonacci numbers are part of a sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21 and so on. As can be seen, each number is the sum of the two numbers before it, after the first two numbers. A general equation can therefore be formed:

The Fibonacci sequence is named after Leonardo of Pisa, an Italian mathematician also known as Fibonacci.

Origins

Fibonacci first thought of the sequence as a solution to a problem he posed in a book he wrote, Liber Abaci. The problem was to find out the number of rabbits produced in a rabbit population which starts from a single newly-born pair. It is assumed that each pair produces another pair every month, they each become productive from the second month onwards and the rabbits never die.

Relation to the Golden Ratio

There are many different ways in which the Fibonacci sequence is related to the Golden Ratio. Firstly, the further you go to the right of the sequence, the more the ratio of one term to the one before it estimates the Golden Ratio. The table below shows the first few numbers and their ratios:

To describe this graphically, we can use squares. Start with a 1 by 1 square. Then, add another square of the same size. Subsequently, add squares whose sides are equal to the longest side of the existing rectangle. After we do this four times, we get a rectangle similar to the following:

The rectangle gets increasingly closer to the golden rectangle, where the ratio of the width to the height is the Golden Ratio. If we draw a quarter circle in each of the squares, we get a Fibonacci spiral as such:

It's one of those things that can be done yet, no - one can do it. In theory you should be able to do it with your bare hands nothing more.

Here's how:

• The thickness of a piece of paper on average is around 0.01 cm thick; let's take an A4 piece of paper for example.
• Firstly, as you may or may not know, a piece of paper cannot be halved more than 7 times, it's a mathematical fact that which cannot be argued against, try if you don't believe me!
• I'm sure that you'll agree when I say that a piece of paper will double in thickness if you fold it in half, hopefully that's pretty obvious
• So lets first consider how large a piece of paper would be if you folded it in half 10 times

0.01x2x2x2x2x2x2x2x2x2 = 10.24cm (try it on a calculator if you need reassurance)

• I'm sure you'll agree - not too thick!
• What about folding the paper 20 times

Well, then it would be a massive 104 metres thick. If you need convincing though, remember that the thickness keeps on doubling so if you did 10.24 x 2 that would be equal to 11 folds not 20.

• So, let's try 30 folds, oh and remember that there are 1000m in one km
• 30 folds = thickness of 107km (Birmingham to Oxford)
• 40 folds = thickness of 68,719 miles (Around the equator 3 times!) (1.6km = 1 mile)
• 50 folds = a massive thickness of 70,369,744 miles (This Sun is 92 million miles away)
• Ready to go interplanetary! 60 folds = an unbelievable thickness of 72 billion miles, regardless of what anyone says, that's a long way to walk!

72 billion miles is the distance to the Sun and back 387 times.

• If you fold a piece of paper 66 times that's one light year!
• A light year is how far a beam of light travels in one year!
• All from one piece of A4 paper imagine if someone could fold a piece of paper more than 7 times!

Use your imagination!