Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur in Plants is a book on the mathematics of plant structure, and in particular on phyllotaxis, the arrangement of leaves on plant stems. It was written by Irving Adler, and published in 2012 by World Scientific. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries. [1]
Irving Adler (1913–2012) was known as a peace protester, schoolteacher, and children's science book author [2] before, in 1961, earning a doctorate in abstract algebra. Even later in his life, Adler began working on phyllotaxis, the mathematical structure of leaves on plant stems. This book, which collects several of his papers on the subject previously published in journals and edited volumes, [3] is the last of his 85 books to be published before his death. [2]
Different plants arrange their leaves differently, for instance on alternating sides of the plant stem, or rotated from each other by other fractions of a full rotation between consecutive leaves. In these patterns, rotations by 1/2 of an angle, 1/3 of an angle, 3/8 of an angle, or 5/8 of an angle are common, and it does not appear to be coincidental that the numerators and denominators of these fractions are all Fibonacci numbers. Higher Fibonacci numbers often appear in the number of spiral arms in the spiraling patterns of sunflower seed heads, or the helical patterns of pineapple cells. [1] The theme of Adler's work in this area, in the papers reproduced in this volume, was to find a mathematical model for plant development that would explain these patterns and the occurrence of the Fibonacci numbers and the golden ratio within them. [4]
The papers are arranged chronologically; they include four journal papers from the 1970s, another from the late 1990s, and a preface and book chapter also from the 1990s. Among them, the first is the longest, and reviewer Adhemar Bultheel calls it "the most fundamental"; it uses the idea of "contact pressure" to cause plant parts to maximize their distance from each other and maintain a consistent angle of divergence from each other, and makes connections with the mathematical theories of circle packing and space-filling curves. Subsequent papers refine this theory, make additional connections for instance to the theory of continued fractions, and provide a more general overview. [4]
Interspersed with the theoretical results in this area are historical asides discussing, among others, the work on phyllotaxis of Theophrastus (the first to study phyllotaxis), Leonardo da Vinci (the first to apply mathematics to phyllotaxis), Johannes Kepler (the first to recognize the importance of the Fibonacci numbers to phyllotaxis), and later naturalists and mathematicians. [1]
Reviewer Peter Ruane found the book gripping, writing that it can be read by a mathematically inclined reader with no background knowledge in phyllotaxis. He suggests, however, that it might be easier to read the papers in the reverse of their chronological order, as the broader overview papers were written later in this sequence. [1] And Yuri V. Rogovchenko calls its publication "a thoughtful tribute to Dr. Adler’s multi-faceted career as a researcher, educator, political activist, and author". [3]
In mathematics, the Fibonacci numbers, commonly denoted Fn , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes from 1 and 2. Starting from 0 and 1, the first few values in the sequence are:
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities and with
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated like a wallpaper design.
A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral and the logarithmic spiral. Fermat spirals are named after Pierre de Fermat.
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider by a factor of φ for every quarter turn it makes.
In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the smaller arc to the length of the larger arc is the same as the ratio of the length of the larger arc to the full circumference of the circle.
In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature.
On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.
Stephen Louis Adler is an American physicist specializing in elementary particles and field theory. He is currently Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton, New Jersey.
As the tip of a plant shoot grows, new leaves are produced at regular time intervals if temperature is held constant. This time interval is termed the plastochron. The plastochrone index and the leaf plastochron index are ways of measuring the age of a plant dependent on morphological traits rather than on chronological age. Use of these indices removes differences caused by germination, developmental differences and exponential growth.
Irving Adler was an American author, mathematician, scientist, political activist, and educator. He was the author of 57 books about mathematics, science, and education, and the co-author of 30 more, for both children and adults. His books have been published in 31 countries in 19 different languages. Since his teenaged years, Adler was involved in social and political activities focused on civil rights, civil liberties, and peace, including his role as a plaintiff in the McCarthy-era case Adler vs. Board of Education that bears his name.
A leaf is any of the principal appendages of a vascular plant stem, usually borne laterally aboveground and specialized for photosynthesis. Leaves are collectively called foliage, as in "autumn foliage", while the leaves, stem, flower, and fruit collectively form the shoot system. In most leaves, the primary photosynthetic tissue is the palisade mesophyll and is located on the upper side of the blade or lamina of the leaf but in some species, including the mature foliage of Eucalyptus, palisade mesophyll is present on both sides and the leaves are said to be isobilateral. Most leaves are flattened and have distinct upper (adaxial) and lower (abaxial) surfaces that differ in color, hairiness, the number of stomata, the amount and structure of epicuticular wax and other features. Leaves are mostly green in color due to the presence of a compound called chlorophyll that is essential for photosynthesis as it absorbs light energy from the sun. A leaf with lighter-colored or white patches or edges is called a variegated leaf.
Miodrag S. Petković is a mathematician and computer scientist. In 1991 he became a full professor of mathematics at the Faculty of Electronic Engineering, University of Niš in Serbia.
Peggy Adler is an American author & illustrator and investigative researcher. She is the daughter of Irving Adler and Ruth Adler and younger sister of Stephen L. Adler.
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.
In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ, ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two. Unless θ is a rational multiple of π, there will also be at least two distinct distances.
Closing the Gap: The Quest to Understand Prime Numbers is a book on prime numbers and prime gaps by Vicky Neale, published in 2017 by the Oxford University Press (ISBN 9780198788287). The Basic Library List Committee of the Mathematical Association of America has suggested that it be included in undergraduate mathematics libraries.
David S. Richeson is an American mathematician whose interests include the topology of dynamical systems, recreational mathematics, and the history of mathematics. He is a professor of mathematics at Dickinson College, where he holds the John J. & Ann Curley Faculty Chair in the Liberal Arts.
Kim Williams is an American architect, an independent scholar on the connections between architecture and mathematics, and a book publisher. She is the founder of the Nexus: Architecture and Mathematics conference series, the founder and co-editor-in-chief of Nexus Network Journal, and the author of several books on mathematics and architecture.
In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane in which each circle is surrounded by a ring of six tangent circles. These patterns contain spiral arms formed by circles linked through opposite points of tangency, with their centers on logarithmic spirals of three different shapes.