This book focuses on the origin of the Gielis curves, surfaces and transformations in the plant sciences. It is shown how these transformations, as a generalization of the Pythagorean Theorem, play an essential role in plant morphology and development. New insights show how plants can be understood as developing mathematical equations, which opens the possibility of directly solving analytically any boundary value problems (stress, diffusion, vibration...) . The book illustrates how form, development and evolution of plants unveil as a musical symphony. The reader will gain insight in how the methods are applicable in many divers scientific and technological fields.
Now available in an affordable softcover edition, this classic in Springer's acclaimed Virtual Laboratory series is the first comprehensive account of the computer simulation of plant development. 150 illustrations, one third of them in colour, vividly demonstrate the spectacular results of the algorithms used to model plant shapes and developmental processes. The latest in computer-generated images allow us to look at plants growing, self-replicating, responding to external factors and even mutating, without becoming entangled in the underlying mathematical formulae involved. The authors place particular emphasis on Lindenmayer systems - a notion conceived by one of the authors, Aristid Lindenmayer, and internationally recognised for its exceptional elegance in modelling biological phenomena. Nonetheless, the two authors take great care to present a survey of alternative methods for plant modelling.
An instant New York Times Bestseller! “Unreasonably entertaining . . . reveals how geometric thinking can allow for everything from fairer American elections to better pandemic planning.” —The New York Times From the New York Times-bestselling author of How Not to Be Wrong—himself a world-class geometer—a far-ranging exploration of the power of geometry, which turns out to help us think better about practically everything. How should a democracy choose its representatives? How can you stop a pandemic from sweeping the world? How do computers learn to play Go, and why is learning Go so much easier for them than learning to read a sentence? Can ancient Greek proportions predict the stock market? (Sorry, no.) What should your kids learn in school if they really want to learn to think? All these are questions about geometry. For real. If you're like most people, geometry is a sterile and dimly remembered exercise you gladly left behind in the dust of ninth grade, along with your braces and active romantic interest in pop singers. If you recall any of it, it's plodding through a series of miniscule steps only to prove some fact about triangles that was obvious to you in the first place. That's not geometry. Okay, it is geometry, but only a tiny part, which has as much to do with geometry in all its flush modern richness as conjugating a verb has to do with a great novel. Shape reveals the geometry underneath some of the most important scientific, political, and philosophical problems we face. Geometry asks: Where are things? Which things are near each other? How can you get from one thing to another thing? Those are important questions. The word "geometry"comes from the Greek for "measuring the world." If anything, that's an undersell. Geometry doesn't just measure the world—it explains it. Shape shows us how.
An exquisite visual celebration of the 2,500-year history of geometry If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configurations involving infinity. The result is a delightful and informative illustrated tour through the 2,500-year-old history of one of the most important branches of mathematics.
The sacred Flower of Life pattern, the primary geometric generator of all physical form, is explored in even more depth in this volume, the second half of the famed Flower of Life workshop. The proportions of the human body, the nuances of human consciousness, the sizes and distances of the stars, planets and moons, even the creations of humankind, are all shown to reflect their origins in this beautiful and divine image. Through an intricate and detailed geometrical mapping, Drunvalo Melchizedek shows how the seemingly simple design of the Flower of Life contains the genesis of our entire third-dimensional existence. From the pyramids and mysteries of Egypt to the new race of Indigo children, Drunvalo presents the sacred geometries of the Reality and the subtle energies that shape our world. We are led through a divinely inspired labyrinth of science and stories, logic and coincidence, on a path of remembering where we come from and the wonder and magic of who we are. Finally, for the first time in print, Drunvalo shares the instructions for the Mer-Ka-Ba meditation, step-by-step techniques for the re-creation of the energy field of the evolved human, which is the key to ascension and the next dimensional world.if done from love, this ancient process of breathing prana opens up for us a world of tantalizing possibility in this dimension, from protective powers to the healing of oneself, of others and even of the planet. Embrace the expanded vision and understanding that Drunvalo offers to the world. Coincidences abound, miracles flourish and the amazing stories of mysteries unveiled arise as the author probes the Ancient Secrets of the Flower of Life.
This enlightening and gorgeously illustrated book explores the beauty and mystery of the divine proportion in art, architecture, nature, and beyond. From the pyramids of Giza, to quasicrystals, to the proportions of the human face, the golden ratio has an infinite capacity to generate shapes with exquisite properties. Author Gary Meisner has spent decades researching the subject, investigating and collaborating with people across the globe in dozens of professions and walks of life. In The Golden Ratio, he shares his enlightening journey. Exploring the long history of this fascinating number, as well as new insights into its power and potential applications, The Golden Ratio invites you to take a new look at this timeless topic.
A fascinating and inspirational look at the vital link between the hidden geometrical order of the universe, geometry in nature, and the geometry of the man-made world. The Da Vinci Code has awakened the public to the powerful and very ancient idea that religious truths and mathematical principles are intimately intertwined. Sacred Geometry offers an accessible way of understanding how that connection is revealed in nature and the arts. Over the centuries, temple builders have relied on magic numbers to shape sacred spaces, astronomers have used geometry to calculate holy seasons, and philosophers have observed the harmony of the universe in the numerical properties of music. By showing how the discoveries of mathematics are manifested over and over again in biology and physics, and how they have inspired the greatest works of art, this illuminating study reveals the universal principles that link us to the infinite.
1-systems are a mathematical formalism which was proposed by Aristid 1indenmayer in 1968 as a foundation for an axiomatic theory of develop ment. The notion promptly attracted the attention of computer scientists, who investigated 1-systems from the viewpoint of formal language theory. This theoretical line of research was pursued very actively in the seventies, resulting in over one thousand publications. A different research direction was taken in 1984 by Alvy Ray Smith, who proposed 1-systems as a tool for synthesizing realistic images of plants and pointed out the relationship between 1-systems and the concept of fractals introduced by Benoit Mandel brot. The work by Smith inspired our studies of the application of 1-systems to computer graphics. Originally, we were interested in two problems: • Can 1-systems be used as a realistic model of plant species found in nature? • Can 1-systems be applied to generate images of a wide class of fractals? It turned out that both questions had affirmative answers. Subsequently we found that 1-systems could be applied to other areas, such as the generation of tilings, reproduction of a geometric art form from East India, and synthesis of musical scores based on an interpretation of fractals. This book collects our results related to the graphical applications of- systems. It is a corrected version of the notes which we prepared for the ACM SIGGRAPH '88 course on fractals.
Second part of a larger project exploring the relationship between mathematics and Nature. The first part focused on the geometry of minerals. This book is a visual journey that celebrates the many connections between the organic world and the findings of abstract mathematical sciences throughout history. 52 illustrations & original photographs. Logbook & references.
