THE BEAUTY OFMATHEMATICS Dr. Farhana Shaheen Assistant ProfessorYanbu University College KSA
Board Maths is Bore Maths?• Is Maths really boring???????????• No mystery; No beauty; No glamour?• Is it really DRY?• Where do you see Maths, other than books?• Was Maths Invented? Discovered? Or…..• Was it “FORMULATED???”…………• From NATURE? By Man?
• Allah created Universe, and gave it a system.• Lunar, Solar, Galaxies and many others.• Man studied that system and then formulized it.• Many theories were evolved, discoveries were made; there were inventions and creations…, man reached to the moon… all because of Mathematics…being the Queen of all Sciences.
All the planets in the Solar Systemrevolve around the Sun, includingour own planet, the Earth.
Mathematics: Comes from theGreek word “Mathema” meaning“Science, Knowledge, or Learning”• The study of Figures and Numbers.• The study of Patterns of Structure, Change and Space.• The language of Sciences• A Science dealing with the Logic of Quantity, Shape and Arrangement.• The Science that deals with Numbers, Quantities, Shapes, Patterns, Measurement and the concepts related to them.
Branches of Mathematics• Foundations: Set Theory; Logic and Model Theory; Category Theory; Computability Theory• Algebra: Group Theory; Ring Theory; Field Theory; Galois Theory; Number Theory; Combinatorics; Arithmetic; Module Theory;• Geometry and Topology: Euclidean and Non-Euclidean Geometry; Metric Geometry; Projective Geometry Differential Geometry; Graph Theory; Applied Mathematics: Statistics; Probability Theory; Computer Science; Game Theory; Systems and Control Theory; Modeling and Simulation; Physics; etc.
Study of Shapes:• In our everyday life, we see geometric figures like circles, squares, triangles, polygons, etc.
The Inspiration of Honey Bees• Honey Bees display an extra ordinary architectural skill. The hexagonal cells they build are based on complex mathematical calculations. They use a system whereby they can do the maximum storage with the minimum material.• The interesting aspect of the cells is that honey bees start to build the cells from different points and meet in the middle, with no discord at the intersection point.
BEAUTY OF NUMBERS• Multiply of 37 by multiplies of 3: 3 x 37 = 111 6 x 37 = 222 9 x 37 = 333 12 x 37 = 444 15 x 37 = 555 18 x 37 = 666 21 x 37 = 777 24 x 37 = 888 27 x 37 = 999
• Trapeze: 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 1111 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111
• Trapeze: 1x8+1=9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432 123456789 x 8 + 9 = 987654321
• 0x9+8 =8 9 x 9 + 7 = 88 98 x 9 + 6 = 888 987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 + 1 = 88888888 98765432 x 9 + 0 = 888888888 987654321 x 9 - 1 = 8888888888 9876543210 x 9 - 2 = 88888888888•
Take a look at this symmetry:• 1x1=1 11 x 11 = 121 111 x 111 = 12321 1111 x 1111 = 1234321 11111 x 11111 = 123454321 111111 x 111111 = 12345654321 1111111 x 1111111 = 1234567654321 11111111 x 11111111 = 123456787654321 111111111 x 111111111 = 12345678987654321
“Fibonacci numbers”• Observe the series of numbers:• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ……• Note that every number is obtained by adding the previous two. e.g. 8 = 5 + 3,• 13 = 8 + 5, and so on.• In general, we have f(n) = f(n-1) + f(n-2).• So, the numbers 1,2,3,5,8,… etc. are are Fibonacci nos. Here we assume f(0) = 0.
Fibonacci Numbers• The Fibonacci Number Sequence was first presented in Leonardo Pisanos book, "Liber abaci" or "Book of Calculating". It is a sequence that I find to be very fascinating, and surprisingly it is a part of every day nature.• The Fibonacci sequence can be found in sea shell spirals, branching plants, petals on flowers, in pine cones and even in human body.
Seeds in fruits are arranged inFibonnaci numbers• Let us look at an apple? Instead of cutting it from the stalk to the opposite end (where the flower was), i.e. from "North pole" to "South pole", try cutting it along the "Equator".
Cutting the apple horizontally, i.e.along the "Equator“…• You „ll find a five petal flower… Fibonacci number.
Fibonacci number in Banana• What about a banana? Count how many "flat" surfaces it is made from - is it 3 or perhaps 5? When youve peeled it, cut it in half (as if breaking it in half, not lengthwise) and look again. Surprise! Theres a Fibonacci number.
Petals of flower are in Fibonaccinumbers • Lily: Although these appear to have 6 petals as shown , 3 are in fact sepals and 3 are petals. Sepals form the outer protection of the flower when in bud.
• The 3 sepals that protected the bud are outermost, then 5 outer green petals followed by an inner layer of 5 more paler green petals
Examples• 4 petals Very few plants show 4 petals (or sepals) but some, such as the fuchsia do. 4 is not a Fibonacci number!• 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia), pinks . The humble buttercup has been bred into a multi-petalled form. 8 petals: delphiniums 13 petals: ragwort, corn marigold, cineraria, some daisies 21 petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, the asteraceae family. Some species are very precise about the number of petals they have - eg buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.
Seed heads• This poppy seed head has 13 ridges on top
Seeds in Sunflower, spirallingoutwards both left and right, are arranged in Fibonnaci Series
Leaf arrangements• Many plants show the Fibonacci numbers in the arrangements of the leaves around their stems. If we look down on a plant, the leaves are often arranged so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.
Leaves per turn• The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one.• If we count in the other direction, we get a different number of turns for the same number of leaves.• The number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers!
• For example, in the top plant in the picture above, we have 3 clockwise rotations before we meet a leaf directly above the first, passing 5 leaves on the way. If we go anti-clockwise, we need only 2 turns. Notice that 2, 3 and 5 are consecutive Fibonacci numbers. For the lower plant in the picture, we have 5 clockwise rotations passing 8 leaves, or just 3 rotations in the anti-clockwise direction. This time 3, 5 and 8 are consecutive numbers in the Fibonacci sequence. We can write this as, for the top plant, 3/5 clockwise rotations per leaf ( or 2/5 for the anticlockwise direction). For the second plant it is 5/8 of a turn per leaf (or 3/8).• The sunflower here when viewed from the top shows the same pattern. It is the same plant whose side view is above. Starting at the leaf marked "X", we find the next lower leaf turning clockwise. Numbering the leaves produces the patterns shown.
• You will see that the third leaf and fifth leaves are next nearest below our starting leaf but the next nearest below it is the 8th then the 13th. How many turns did it take to reach each leaf? Leaf number:3 5 8• Turns Clockwise: 1 2 3• The pattern continues with Fibonacci numbers in each column! Leaf arrangements of some common plants• One estimate is that 90 percent of all plants exhibit this pattern of leaves involving the Fibonacci numbers.• Some common trees with their Fibonacci leaf arrangement numbers are:• 1/2 elm, linden, lime, grasses 1/3 beech, hazel, grasses, blackberry 2/5 oak, cherry, apple, holly, plum, common groundsel 3/8 poplar, rose, pear, willow 5/13 pussy willow, almond where t/n means each leaf is t/n of a turn after the last leaf or that there is there are t turns for n leaves.
The Golden Ratio: Phi• The ratio of 1:1.61803 has unique and storied history. It is also know as Phi, named after the Greek sculptor Phidias. The ratio is found in nature, art, architecture, poetry, music, and of course, in Mathematics.• Phidias incorporated the ratio into many of his sculptures, one of which is the ancient Greek temple, the Parthenon. This was a rectangle shaped temple whose sides are in the golden proportion.
Golden ratio• Two quantities are said to be in the golden ratio, if "the whole is to the larger as the larger is to the smaller“.•
Fibonacci numbers follow Golden Ratio found in Golden Rectangle• The Golden ratio phi =1.61803…… has dazzled mathematicians due to its strange and inspiring properties. It appears every where, from Egyptian pyramids to Renaissance art.• Phi emerges in a rectangle which has sides in a:b ratio. The Fibonacci seq.• 1/1, 2/1, 3/2, 5/3, 8/5, 13/8,…….leads to the Golden ratio phi.
Equiangular Spirals or Natilus Shell The spiral inside the shell is close to the golden ratio
ART & GOLDEN RATIO• Throughout the centuries, artists and musicians - pursuing aesthetics - have turned to nature‟s own tactics and emulated the golden ratio in their own creations.• The renaissance was a period of renewed interest and advancement in the arts. It turns out that some of the most famous artists of all time had a very mathematical approach to their work.
• Leonardo da Vinci was a mathematician and a scientist as much as he was an artist: along with other techniques he frequently used the divine proportion in his paintings.
• GRAVEYARD CROSSES A German psychologist by the name of Gustav Fechner studied the crosses in graveyards and discovered that the ratio of the lengths of the upper and lower portions of the main stem is in golden proportion too.
The golden proportion is clearlyevident in the human figure:• Ones upper portion of leg can often be found to be Phi times ones lower portion;• Ones eyes are usually about Phi of the way up from the chin.
The proportion of distance betweenthe two eyes to size of an eye is alsophi (a set gauge is being used tomeasure the teeth).
The proportion of adjacent teeth isalso phi (a set gauge is being usedto measure the teeth).
• If you measure the lengths of the bones in your finger (best seen by slightly bending the finger) does it look as if the ratio of the longest bone in a finger to the middle bone is Phi? What about the ratio of the middle bone to the shortest bone (at the end of the finger) - Phi again?
MATHEMATICS THE QUEEN OF ALL SCIENCES• So……………….who says MATHS IS…• BORING???• DRY???• UNINTERESTING???• NO BEAUTY???• NO GLAMOUR???• DIFFICULT???• Well, I believe… NOTHING IS DIFFICULT ONLY IF YOU KNOW HOW.• THANK YOU
If you are kind, people mayAccuse you of being selfish,And having ulterior motives; Be kind anyway.