Patterns in Nature Paul Harrington and Juliana Kew Contents Ratios in the Human Body………………………………..3 The Vitruvian Man………………………………………4 The Fibonacci Sequence………………………………….5 Fibonacci Sequence in Bees……………………………….6 Fibonacci Sequence in Rabbits……………………………8 Fibonacci Sequence in Flowers……………………………9 Spirals in Nature………………………………………..10 The Fibonacci Spiral……………………………………11 Fractals…………………………………………………12 Fractals in Bacteria Colonies……………………………..13 Fractals in Vegetables……………………………………14 Fractal Electric Discharge………………………………..15 Simple Natural Fractals…………………………………16 Tessellations……………………………………………17 Tessellations in Honeycomb……………………………..18 Tessellations in Rock……………………………………19 Glossary………………………………………………..20 Image Credits…………………………………………...21 About the Authors……………………………………...22 Ratios in the Human Body Cartoons Small children have disproportionately large heads, which is part of what distinguishes them as cute. A baby’s head grows as its body does, but not as quickly or as much. Eventually the body catches up to the scale of the head. Some ratios appear often in the study of human anatomy. Human beings are highly variable, and so the ratios that describe the average or ideal individual may not hold true for everyone. However, some are often accurate. Cartoon style toys and drawings take advantage of disproportionality to be more appealing. height : arm span :: 1 : 1 height : femur : : 4 : 1 height : head circumference : : 3 : 1 height : head height : : 8 : 1 foot length : palm width : : 4 : 1 face length : ear length : : 3 : 1 The Vitruvian Man The ratios of the human body are often observed in art, where regular proportions seem more natural or beautiful to the viewer. The most famous study of human proportion is Leonardo DaVinci’s Vitruvian Man. The name comes from the ancient Roman architect Vitruvius, whose writings on human proportion DaVinci referenced. The drawing shows that the man standing with feet together and arms wide can be inscribed in a square, which makes sense if his height equals his arm span. The second image of the man, spread-eagled this time, is inscribed in a circle. The Fibonacci Sequence The Fibonacci sequence was discovered by Leonardo of Pisa, who was called Fibonacci (a contraction of filius Bonacci, Latin for “son of Bonacci”). He introduced his work to western mathematics in the book Liber Abaci, even though the pattern had been described by earlier Indian mathematicians. The pattern begins with two numbers, zero and one. Other than the first two, any number in the sequence is the sum of the two numbers before it. Some simple calculations yield the following series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. This method of generating the sequence is known as a recursively defined function. The two numbers are defined (F[0] = 0 and F[1] = 1) and the rest are defined by the previous numbers (F[n] = F[n-1] + F[n-2]). Fibonacci Sequence in Bees Bee birthing follows two simple rules. A male bee hatches from an unfertilized egg, and a female bee hatches from a fertilized egg. A single female can lay an unfertilized egg, but a male and female can lay a fertilized egg. While following the bees related to a single male, we find the Fibonacci sequence. We have to assume that all the parents are not related. First, we have the single male bee (1). Then, we have its female parent (1). The female has two parents: a male and a female (2). The male grandparent has a one parent, and the female has two: one male, two females (3). The females are the offspring of two bees each, and the male is the offspring of one: two males, three females (5). Placing the numbers next to one another, we have 1, 1, 2, 3, and 5, which are the first five elements of the Fibonacci sequence. Fibonacci Sequence in Bees (cont.) Following the ancestry of a female, a similar pattern occurs. We start with the single female (1). It has two parents, a male and a female (2). There are three grandparents: one male, two females (3). There are five great grandparents: two males, three females (5). Placing the numbers next to one another, we have 1, 2, 3, and 5, the second Fibonacci number to the fifth. Fibonacci Sequence in Rabbits Rabbit mating pairs follow a simple pattern. For the first cycle that they exist, they bear no children. For each cycle after that, they have one pair. Assuming that rabbits never die, and that the population can grow without bound, the Fibonacci sequence is found in the number of pairs of rabbits. At first, there is one new pair of rabbits (1). In the next cycle, they mature (1). Now the mature rabbits have one pair of offspring (2). After another cycle, the mature pair has another set of kids, and each of the rabbits from the immature pair become adults (3). Now there are two mature pairs and one immature pair. In the next round, there will be three mature pairs and two immature pairs (5). Placing these numbers next to one another, we have 1, 1, 2, 3, and 5, the first five numbers in the Fibonacci sequence. Fibonacci Sequence in Flowers The Fibonacci series can be found in flowers. A good place to look is the number of petals. Unfortunately, the smallest numbers of the sequence (mainly zero, one, two, and three) are not easily found. However, the larger numbers, such as five and up, are fairly commonplace. The columbine flower has five petals. Bloodroot has eight petals. Black eyed susan has thirteen petals. Various types of daisies have 13, 21, 34, 55, or even 89 petals. Of course, saying that a flower has a number of petals is measure of the mode, or the most common occurrence for that flower. It is possible for a flower to over develop, or more commonly, under develop. For example, black eyed susans can have anywhere between eleven and fifteen petals. Spirals in Nature Spirals can be found in many natural formations. Many different shells have the shape of an Archimedean spiral. Certain galaxies, such as our own Milky Way, also have the same form. Deoxyribonucleic acid, also known as DNA, forms a helical structure, which is a three dimensional spiral, inside the nucleus of cells. Helixes can also be seen in goat horns or climber plant tendrils. Spiral growth refers to an organism’s growing into a spiral shape. It uses the maximum amount of area within a certain radius of the center of the growth. For plants, that maximizes the surface area that is exposed to sunlight. The Fibonacci Spiral A Fibonacci spiral is a shape built from the relationships between Fibonacci numbers. Imagine a square with unit-long sides. To the right place another unit square. Now beneath the two put a third square whose side length is the sum of the lengths of the previous two square side lengths. It will have sides of length 2. Next go to the left and repeat the process, for a square with side lengths of 3. Then move above for a 5x5 square. The pattern can continue indefinitely, right, down, left, up, always adding the side lengths of the previous two squares. To draw the spiral, inscribe a quarter circle in each square. Fractals A fractal is a pattern within a pattern. At the largest level, it looks like a single shape or a group of shapes. But when the fractal is expanded, it can be seen that it is made up of more of the same shape. And each of those shapes is made of even smaller ones. This pattern continues forever, so it is impossible to draw a fractal completely correctly. However, computers can draw fractals with extreme accuracy down to a scale smaller than the human eye can see. Fractals in Bacteria Colonies Usually when colonies of bacteria expand, they form compact shapes, as seen to the left above. However, when the growing colony is stressed, when it does not have a soft base or enough nutrients, it grows into fractal branches. This allows it to expand quickly without extra effort. All the branches give it more surface area further away from the center. At the tips of the branches the bacteria have access to more nutrients, so the colony can grow larger and stay healthy. Fractals in Vegetables Evolution Plants tend to follow the Fibonacci Sequence because it has useful geometric properties. Spirals are a good way to pack many units closely together, and the sequence lends itself easily to spirals. Plants that are arranged according to the Fibonacci Sequence therefore have an advantage over those that are not and over generations are more successful. The more successful plants are the ones that can survive better and so become more common. Romanesco broccoli is an extreme example of a natural fractal, with a basic shape reminiscent of the Fibonacci Sequence. Its underlying structure is a simple cone, but overlaid on that is a spiral of new cones defined by the ratios of Fibonacci numbers. Each of those cones is itself a spiral of smaller cones, and the pattern continues down to a minute scale. This selfsimilarity is what defines the whole vegetable as a natural fractal. Fractal Electric Discharge Electric discharge often forms fractals. When the electricity enters a medium, it is said to take the path of least resistance. However, it doesn’t take just one path. Instead, it travels along multiple pathways. If its medium is uniform throughout, the electricity takes multiple paths in the shape of a fork. Fractals form because the discharge, after forking, reacts to the medium in the same way, but now with a lesser charge. So, it forks again, but each leaf of the fork is smaller than the original fork. As this continues, the fractal becomes more and more detailed. Simple Natural Fractals Landscapes, when viewed from a large distance, can be seen as warped fractals. Mountains, for example, are warped examples of the Sierpinski gasket. Each vertex in the fractal has been slightly moved from its true position, but the method of generating the fractal remained the same. River erosion patterns can form fractals. As the water runs downhill, it splits into several smaller flows. Those flows also split, and so on. Each step of the pattern is a smaller version of the step before it, which makes it a fractal. The image below is a landscape computer-generated using fractals. Tessellations Development Living organisms are vastly complicated assemblies, with special structures called tissues to perform different tasks. For instance, bone tissue is specialized to provide support and cannot contract like muscle tissue can. But both muscles and bones, along with all the other structures in an organism, came originally from the same cells. They diverged and became specialized because during their development different chemicals were secreted in the fluid around them. In response, the cells marshaled themselves into the forms seen in the mature organism. A tessellation is a pattern that completely fills a surface with no gaps or overlaps. Bricks are a simple type of man-made tessellation, but these patterns can also be found in nature. Turtle shells are tessellations; they have hexagons or other similar shapes instead of rectangles, but the tiles cover the whole surface of the shell. The individual tiles are actually outgrowths of the ribs and vertebrae of the turtle, which is why the reptiles cannot leave their shells. It also means that the tiles must butt firmly against each other to lend the shell strength. Without such tessellation, the shell would collapse into the component parts and the turtle would be left defenseless. Tessellations in Honeycomb Honeycomb is a natural tessellation and a very efficient arrangement for storage. Bees build honeycomb with wax, which they produce at a rate of about 1 pound for every 8.4 pounds of honey consumed. With so little wax, it is important that the cells of the honeycomb use as little of it as possible to cover the area of the hive. This is where the hexagons of the comb are useful; hexagons cover the hive with less material than quadrilaterals or triangles. Those three shapes (hexagons, quadrilaterals, and triangles) are the only ones that can completely cover a plane by themselves with no overlap. And hexagons contain the most area with the least perimeter, so they are what bees use to store their honey and larvae. Tessellations in Rock Large areas of rocks can sometimes form tessellations. When lava flows out of a volcano, it covers the ground with a continuous sheet of molten rock. The rock starts to stick to anything underneath it, so it is anchored to the ground as well as to itself. However, it then begins to cool. Like most materials, rock contracts when it cools. But in this case, it cannot just shrink because it is stuck to the ground. Instead, it cracks into hexagonal segments, like the ones seen in the Giant’s Causeway in Northern Ireland. Other kinds of rock can crack into tessellations, too, even if they have not come from lava. An example is the Tessellated Pavement in Tanzania. Glossary Ancestry Colony Contract Cycle Discharge Disproportionately Diverge Femur Inscribe Medium Petal Proportion Quadrilateral Ratio Recursive Secrete Sierpinski Gasket a series of parents and older generations visible cluster of bacteria , theoretically descended from a single cell. decrease in size any complete event that repeats to rid of an electric charge in a ratio different from normal develop in a different direction bone of the thigh draw a figure within another so that their boundaries touch but do not intersect the substance on which something occurs one of the colored segments that surrounds the head of a flower The comparative measurements or size of different parts of a whole 4-sided shape proportion or fraction a description of a cycle that depends on previous cycles produce and discharge fractal formed by dividing a triangle into 4 equal segments Image Credits Cover: http://www.funnyphotos.net.au/fractal/ http://80.33.141.76/pashmina_models/index.php?option=com_content&view=section&layout=blog&id=16&Itemid=27 http://psychedelicadventure.blogspot.com/2009_06_01_archive.html http://www.bergoiata.org/fe/space5/jw%20Year%20V%20Space%20Shots%20056%20-%20Galaxy.jpg http://www.graphicmania.net/wp-content/uploads/crystal-patterns-580739-ga.jpg http://blog.alwaysquiltingonline.com/wp-content/uploads/2009/06/nature-pattern1gif1.jpg http://img.izismile.com/img/img2/20091217/patterns_in_nature_00.jpg http://www.fourmilab.ch/images/Romanesco/images/Scr6.jpg http://i716.photobucket.com/albums/ww165/erhards22/flame.jpg http://scienceblogs.com/startswithabang/upload/2010/04/volcanic_lightning_eyjafjallaj/lightning-1.jpeg http://www.miqel.com/images_1/fractal_math_patterns/natural-patterns/vonkarman_clouds_1.jpg http://brainjabber.wikispaces.com/file/view/spiral-galaxy-wallpaper.jpg/130626239/spiral-galaxy-wallpaper.jpg http://www.freepspwallpapers.net/psp-wallpapers/1/3D/Water-Vortex.jpg http://www.redfieldplugins.com/filterFractalius.htm Contents: http://brainjabber.wikispaces.com/file/view/spiral-galaxy-wallpaper.jpg/130626239/spiral-galaxy-wallpaper.jpg Ratios in the Human Body: http://img107.imageshack.us/img107/8147/girl2prefectmw3.jpg http://www.seanmichaelragan.com/html/%5B2007-12-23%5D_LEGO_mecha_minifig_head_exploit.shtml http://soniceview.blogspot.com/2010/10/chibi.html The Vitruvian Man: http://thequickglimpse.wordpress.com/2010/02/12/vitruvian-man-x-rayed/ The Fibonacci Sequence: http://mrsvesseymathematicians.wikispaces.com/Fibonacci https://gcdxy.wordpress.com/2011/02/07/fibonacci-numbers-the-golden-spiral/ Fibonacci Sequence in Bees: http://www.chabad.org/library/article_cdo/aid/463900/jewish/Deciphering-Natures-Code.htm Fibonacci Sequence in Rabbits: http://www.landlearn.net.au/newsletter/2008term3/images/rabbit-family-tree.png Fibonacci Sequence in Flowers: http://flowerinfo.org/wp-content/gallery/columbine-flowers/columbine-flower-3.jpg http://www.earthhealing.info/bloodroot.jpg http://www.spincerelyyours.com/wp-content/uploads/2007/07/black-eyed-susan.jpg Spirals in Nature: http://originalbeauty.files.wordpress.com/2009/06/400px-goat_with_spiral_horns.jpg?w=400&h=600 http://lh6.ggpht.com/_07ZDmVR-uzM/TCAkcrXbclI/AAAAAAAAAKY/aun3EeA8xwg/Ammonite-5-1024.jpg Fibonacci Spiral: http://dooleymath.com/Algebra/FibonacciSpiral.html Fractals: http://egregores.blogspot.com/2010/12/extremely-cool-natural-fractals.html http://www.fractal-recursions.com/files/fractal-12060301.html Fractals in Bacteria Colonies: http://www.temple.edu/dentistry/admissions/course_descriptions.html http://biocurious.com/images/fractalBacteria.jpg http://www.jessicasachs.com/blog/2006/03/ben-jacobs-fractal-bacteria.html Fractals in Vegetables: http://scienceblogs.com/chaoticutopia/2006/11/friday_fractal_xxv.php http://www.growinghappiness.com/tag/pms/ Fractal Electric Discharge: http://pixdaus.com/pics/1240446086AeJn4eA.jpg http://www.miqel.com/images_1/fractal_math_patterns/natural-patterns/fractal_cd_microwave.jpg http://www.miqel.com/images_1/fractal_math_patterns/natural-patterns/lichtenberg_figure.jpg Simple Natural Fractals: http://www.miqel.com/images_1/fractal_math_patterns/natural-patterns/FractalLandscapebytheOstrich.jpg http://www.cnsm.csulb.edu/departments/geology/people/bperry/GrantPhotos/PtCon3/014ErosionCalienteRangeOct06S.jpg Tessellations: http://britton.disted.camosun.bc.ca/jbtess97.htm http://www.i-club.com/forums/showthread.php?t=188886 http://picfind.bloguez.com/picfind/1653718/turtle-shell Tessellations in Honeycomb: http://thundafunda.com/33/travel-world-pictures/Honeycomb.php Tessellations in Rock: http://www.great-britain.co.uk/world-heritage/giants-causeway.htm http://pix.alaporte.net/pub/Australia/Tasmania/Tasman+Peninsula/ About the Authors: http://www.couchcampus.com/television/tv-talk/top-twenty-tv-series-on-hulu/ http://www.411mania.com/siteimages/metaknight_83017.jpg http://media.wizards.com/images/magic/daily/wallpapers/wp_2012planeswalkers_1280x960.jpg http://sabrinamina.wordpress.com/author/sabrinamina/ http://capecodhistory.us/quotes/politics.htm About the Authors Paul Harrington is a junior at the Massachusetts Academy of Math and Science. He lives in Peabody, MA. Paul can often be found playing Super Smash Brothers Brawl as his favorite character, Metaknight. Also, Paul is a fan of Magic: The Gathering, which he plays almost as often as SSBB. Julie Kew is a high school student at the Mass Academy of Math and Science. She enjoys languages, mythology, and connecting everything possible to astronautics.