László Fejes Tóth
László Fejes Tóth was a Hungarian mathematician who specialized in geometry. He proved that a lattice pattern is the most efficient way to pack centrally symmetric convex sets on the Euclidean plane, he investigated the sphere packing problem. He was the first to show, in 1953, that proof of the Kepler conjecture can be reduced to a finite case analysis and that the problem might be solved using a computer, he was a member of the Hungarian Academy of Sciences and a director of the Alfréd Rényi Institute of Mathematics. He received both State Award. Together with H. S. M. Coxeter and Paul Erdős, he laid the foundations of discrete geometry; as described in a 1999 interview with István Hargittai, Fejes Tóth’s father was a railway worker, who advanced in his career within the railway organization to earn a doctorate in law. Fejes Tóth's mother taught German literature in a high school; the family moved to Budapest. Fejes Tóth attended Pázmány Péter University, now the Eötvös Loránd University; as a freshman, he developed a generalized solution regarding Cauchy exponential series, which he published in the proceedings of the French Academy of Sciences—1935.
He received his doctorate at Pázmány Péter University, under the direction of Lipót Fejér. After university, he received a medical exemption. In 1941 he joined the University of Kolozsvár, it was here. In 1944, he returned to Budapest to teach mathematics at Árpád High School. Between 1946 and 1949 he lectured at Pázmány Péter University and starting in 1949 became a professor at the University of Veszprém for 15 years, where he was the primary developer of the "geometric patterns" theory "of the plane, the sphere and the surface space" and where he "had studied non grid-like structures and quasicrystals" which became an independent discipline, as reported by János Pach; the editors of a book dedicated to Fejes Tóth described some highlights of his early work. He showed that, of all convex polytopes of given surface area that are equivalent to a given Platonic solid, a regular polytope always has the largest possible volume, he developed a technique that proved Steiner's conjecture for the dodecahedron.
By 1953, Fejes Tóth had written dozens of papers devoted to these types of fundamental issues. His distinguished academic career allowed him to travel abroad beyond the Iron Curtain to attend international conferences and teach at various universities, including those at Freiburg. Fejes Tóth met his wife in university, she was a chemist. They were parents of three children, two sons—one a professor of mathematics at the Alfréd Rényi Institute of Mathematics, the other a professor of physiology at Dartmouth College—and one daughter, a psychologist, he enjoyed sports, being skilled at table tennis and gymnastics. A family photograph shows him swinging by his arms over the top of a high bar when he was around fifty. Fejes Tóth held the following positions over his career: Assistant instructor, University of Kolozsvár Teacher, Árpád High School Private Lecturer, Pázmány Péter University Professor, University of Veszprém Researcher director, Mathematical Research Institute In addition to his positions in residence, he was a corresponding member of the Saxonian Academy of Sciences and Humanities, Akademie der Wissenschaften der DDR, of the Braunschweigische Wissenschaftlische Gesellschaft.
According to J. A. Todd, a reviewer of Fejes Tóth's book Regular Figures, Fejes Tóth divided the topic into two sections. One, entitled "Systematology of the Regular Figures", develops a theory of "regular and Archimedean polyhedra and of regular polytopes". Todd explains that the treatment includes: Plane Ornaments, including two-dimensional crystallographic groups Spherical arrangements, including an enumeration of the 32 crystal classes Hyperbolic tessellations, those discrete groups generated by two operations whose product is involutary Polyhedra, including regular solids and convex Archimedean solids Regular polytopes The other section, entitled "Genetics of the Regular Figures", covers a number of special problems, according to Todd; these problems include "packings and coverings of circles in a plane, and... with tessellations on a sphere" and problems "in the hyperbolic plane, in Euclidean space of three or more dimensions." At the time, Todd opined that those problems were "a subject in which there is still much scope for research, one which calls for considerable ingenuity in approaching its problems".
Imre Bárány credited Fejes Tóth with several influential proofs in the field of discrete and convex geometry, pertaining to packings and coverings by circles, to convex sets in a plane and to packings and coverings in higher dimensions, including the first correct proof of Thue's theorem. He credits Fejes Tóth, along with Paul Erdős, as having helped to "create the school of Hungarian discrete geometry."Fejes Tóth's monograph, Lagerungen in der Ebene, auf der Kugel und im Raum, translated into Russian and J
A modem is a hardware device that converts data between transmission media so that it can be transmitted from computer to computer. The goal is to produce a signal that can be transmitted and decoded to reproduce the original digital data. Modems can be used with any means of transmitting analog signals from light-emitting diodes to radio. A common type of modem is one that turns the digital data of a computer into modulated electrical signal for transmission over telephone lines and demodulated by another modem at the receiver side to recover the digital data. Modems are classified by the maximum amount of data they can send in a given unit of time expressed in bits per second or bytes per second. Modems can be classified by their symbol rate, measured in baud; the baud unit denotes symbols per second, or the number of times per second the modem sends a new signal. For example, the ITU V.21 standard used audio frequency-shift keying with two possible frequencies, corresponding to two distinct symbols, to carry 300 bits per second using 300 baud.
By contrast, the original ITU V.22 standard, which could transmit and receive four distinct symbols, transmitted 1,200 bits by sending 600 symbols per second using phase-shift keying News wire services in the 1920s used multiplex devices that satisfied the definition of a modem. However, the modem function was incidental to the multiplexing function, so they are not included in the history of modems. Modems grew out of the need to connect teleprinters over ordinary phone lines instead of the more expensive leased lines, used for current loop–based teleprinters and automated telegraphs. In 1941, the Allies developed a voice encryption system called SIGSALY which used a vocoder to digitize speech encrypted the speech with one-time pad and encoded the digital data as tones using frequency shift keying. Mass-produced modems in the United States began as part of the SAGE air-defense system in 1958, connecting terminals at various airbases, radar sites, command-and-control centers to the SAGE director centers scattered around the United States and Canada.
SAGE modems were described by AT&T's Bell Labs as conforming to their newly published Bell 101 dataset standard. While they ran on dedicated telephone lines, the devices at each end were no different from commercial acoustically coupled Bell 101, 110 baud modems; the 201A and 201B Data-Phones were synchronous modems using two-bit-per-baud phase-shift keying. The 201A operated half-duplex at 2,000 bit/s over normal phone lines, while the 201B provided full duplex 2,400 bit/s service on four-wire leased lines, the send and receive channels each running on their own set of two wires; the famous Bell 103A dataset standard was introduced by AT&T in 1962. It provided full-duplex service at 300 bit/s over normal phone lines. Frequency-shift keying was used, with the call originator transmitting at 1,070 or 1,270 Hz and the answering modem transmitting at 2,025 or 2,225 Hz; the available 103A2 gave an important boost to the use of remote low-speed terminals such as the Teletype Model 33 ASR and KSR, the IBM 2741.
AT&T reduced modem costs by introducing the answer-only 113B/C modems. For many years, the Bell System maintained a monopoly on the use of its phone lines and what devices could be connected to them. However, the FCC's seminal Carterfone Decision of 1968, the FCC concluded that electronic devices could be connected to the telephone system as long as they used an acoustic coupler. Since most handsets were supplied by Western Electric and thus of a standard design, acoustic couplers were easy to build. Acoustically coupled Bell 103A-compatible 300 bit/s modems were common during the 1970s. Well-known models included the Novation CAT and the Anderson-Jacobson, the latter spun off from an in-house project at Stanford Research Institute. An lower-cost option was the Pennywhistle modem, designed to be built using parts from electronics scrap and surplus stores. In December 1972, Vadic introduced the VA3400, notable for full-duplex operation at 1,200 bit/s over the phone network. Like the 103A, it used different frequency bands for receive.
In November 1976, AT&T introduced the 212A modem to compete with Vadic. It used the lower frequency set for transmission. One could use the 212A with a 103A modem at 300 bit/s. According to Vadic, the change in frequency assignments made the 212 intentionally incompatible with acoustic coupling, thereby locking out many potential modem manufacturers. In 1977, Vadic responded with the VA3467 triple modem, an answer-only modem sold to computer center operators that supported Vadic's 1,200-bit/s mode, AT&T's 212A mode, 103A operation; the Hush-a-Phone decision applied only to mechanical connections, but the Carterfone decision of 1968, led to the FCC introducing a rule setting stringent AT&T-designed tests for electronically coupling a device to the phone lines. This opened the door to direct-connect modems that plugged directly into the phone line rather than via a handset. However, the cost of passing the tests was considerable, acoustically coupled modems remained common into the early 1980s.
The falling prices of electronics in the late 1970s led to an increasing number of direct-connect models around 1980. In spite of being directly connected, these modems were operated like their earlier acoustic versions – dialing and other phone-control operations were completed by hand, using an attached handset
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (. Sometimes referred to as the Princeps mathematicorum and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, is ranked among history's most influential mathematicians. Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, to poor, working-class parents, his mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension. Gauss solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years, he was christened and confirmed in a church near the school he attended as a child. Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was three years old he corrected a math error his father made. Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100.
There are many other anecdotes about his precocity while a toddler, he made his first groundbreaking mathematical discoveries while still a teenager. He completed his magnum opus, Disquisitiones Arithmeticae, in 1798, at the age of 21—though it was not published until 1801; this work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day. Gauss's intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum, which he attended from 1792 to 1795, to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems, his breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2. This was a major discovery in an important field of mathematics. Gauss was so pleased with this result that he requested that a regular heptadecagon be inscribed on his tombstone.
The stonemason declined, stating that the difficult construction would look like a circle. The year 1796 was productive for both Gauss and number theory, he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic simplifying manipulations in number theory. On 8 April he became the first to prove the quadratic reciprocity law; this remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and jotted down in his diary the note: "ΕΥΡΗΚΑ! num = Δ + Δ' + Δ". On 1 October he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years led to the Weil conjectures. Gauss remained mentally active into his old age while suffering from gout and general unhappiness.
For example, at the age of 62, he taught himself Russian. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became an associated member of the Royal Institute of the Netherlands. In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen". On the way home from Riemann's lecture, Weber reported that Gauss was full of excitement. On 23 February 1855, Gauss died of a heart attack in Göttingen. Two people gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald, Wolfgang Sartorius von Waltershausen, Gauss's close friend and biographer. Gauss's brain was preserved and was studied by Rudolf Wagner, who found its mass to be above average, at 1,492 grams, the cerebral area equal to 219,588 square millimeters.
Developed convolutions were found, which in the early 20th century were suggested as the explanation of his genius. Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen. Potential evidence that Gauss believed in God comes from his response after solving a problem that had defeated him: "Finally, two days ago, I succeeded—not on account of my hard efforts, but by th
A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about 8.4 lb of honey to secrete 1 lb of wax, so it makes economic sense to return the wax to the hive after harvesting the honey. The structure of the comb may be left intact when honey is extracted from it by uncapping and spinning in a centrifugal machine—the honey extractor. If the honeycomb is too worn out, the wax can be reused in a number of ways, including making sheets of comb foundation with hexagonal pattern; such foundation sheets allow the bees to build the comb with less effort, the hexagonal pattern of worker-sized cell bases discourages the bees from building the larger drone cells. Fresh, new comb is sometimes sold and used intact as comb honey if the honey is being spread on bread rather than used in cooking or as a sweetener. Broodcomb becomes dark over time, because of the cocoons embedded in the cells and the tracking of many feet, called travel stain by beekeepers when seen on frames of comb honey.
Honeycomb in the "supers" that are not used for brood stays light-colored. Numerous wasps Polistinae and Vespinae, construct hexagonal prism-packed combs made of paper instead of wax. However, the term "honeycomb" is not used for such structures; the axes of honeycomb cells are always quasihorizontal, the nonangled rows of honeycomb cells are always horizontally aligned. Thus, each cell has two vertically oriented walls, with the upper and lower parts of the cells composed of two angled walls; the open end of a cell is referred to as the top of the cell, while the opposite end is called the bottom. The cells slope upwards, between 9 and 14°, towards the open ends. Two possible explanations exist as to why honeycomb is composed of hexagons, rather than any other shape. First, the hexagonal tiling creates a partition with equal-sized cells, while minimizing the total perimeter of the cells. Known in geometry as the honeycomb conjecture, this was given by Jan Brożek and proved much by Thomas Hales.
Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume. A second reason, given by D'Arcy Wentworth Thompson, is that the shape results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this, he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency; the closed ends of the honeycomb cells are an example of geometric efficiency, albeit three-dimensional. The ends are trihedral sections of rhombic dodecahedra, with the dihedral angles of all adjacent surfaces measuring 120°, the angle that minimizes surface area for a given volume; the shape of the cells is such that two opposing honeycomb layers nest into each other, with each facet of the closed ends being shared by opposing cells. Individual cells do not show this geometric perfection: in a regular comb, deviations of a few percent from the "perfect" hexagonal shape occur.
In transition zones between the larger cells of drone comb and the smaller cells of worker comb, or when the bees encounter obstacles, the shapes are distorted. Cells are angled up about 13° from horizontal to prevent honey from dripping out. In 1965, László Fejes Tóth discovered the trihedral pyramidal shape used by the honeybee is not the theoretically optimal three-dimensional geometry. A cell end composed of two hexagons and two smaller rhombuses would be.035% more efficient. This difference is too minute to measure on an actual honeycomb, irrelevant to the hive economy in terms of efficient use of wax, considering wild comb varies from any mathematical notion of "ideal" geometry. Bees used their antennae and legs to manipulate the wax during comb construction, while warming the wax. During the construction of hexagonal cells, wax temperature is between 33.6 and 37.6°C, well below the 40°C temperature at which wax is assumed to be liquid for initiating new comb construction. The body temperature of bees is a factor for regulating an ideal wax temperature for building the comb.
Honeycomb structure Wax foundation Hive frame Jan Dzierzon
Robert J. Lang
Robert J. Lang is an American physicist, one of the foremost origami artists and theorists in the world, he is known for his complex and elegant designs, most notably of animals. He has used computers to study the theories behind origami, he has made great advances in making real-world applications of origami to engineering problems. Lang was born in Dayton and grew up in Atlanta, Georgia. Lang studied electrical engineering at the California Institute of Technology, where he met his wife-to-be, Diane, he earned a master's degree in electrical engineering at Stanford University in 1983, returned to Caltech for a Ph. D. in applied physics, with a dissertation titled Semiconductor Lasers: New Geometries and Spectral Properties. Lang began work for NASA's Jet Propulsion Laboratory in 1988. Lang worked as a research scientist for Spectra Diode Labs of San Jose, at JDS Uniphase of San Jose. Lang has authored or co-authored over 80 publications on semiconductor lasers and integrated optoelectronics, holds 46 patents in these fields.
In 2001, Lang left the engineering field to be consultant. However, he still maintains ties to his physics background: he was the editor-in-chief of the IEEE Journal of Quantum Electronics from 2007 to 2010, has done part-time laser consulting for Cypress Semiconductor, among others. Lang resides in Alamo, California. Lang was introduced to origami at the age of six by a teacher who had exhausted other methods of keeping him entertained in the classroom. By his early teens, he was designing original origami patterns. Lang used origami as an escape from the pressures of undergraduate studies. While studying at Caltech, Lang came into contact with other origami masters such as Michael LaFosse, John Montroll, Joseph Wu, Paul Jackson through the Origami Center of America, now known as OrigamiUSA. While in Germany for postdoctoral work and his wife were enamored of Black Forest cuckoo clocks, he became a sensation in the origami world when he folded one after three months of design and six hours of actual folding.
Lang takes full advantage of modern technology in his origami, including using a laser cutter to help score paper for complex folds. Lang is recognized as one of the leading theorists of the mathematics of origami, he has developed ways to algorithmetize the design process for origami, is the author of the proof of the completeness of the Huzita–Hatori axioms. Lang specializes in finding real-world applications for the various theories of origami he has developed; these included designing folding patterns for a German airbag manufacturer. He has worked with the Lawrence Livermore National Laboratory in Livermore, where a team is developing a powerful space telescope, with a 100 m lens in the form of a thin membrane. Lang was engaged by the team to develop a way to fit the tremendous lens, known as the Eyeglass, into a small rocket in such a way that the lens can be unfolded in space and will not suffer from any permanent marks or creases. Lang is many articles on origami. Lang designed the Google Doodle for Akira Yoshizawa's 101st birthday, used by Google on March 14, 2012.
In 2012 he became a fellow of the American Mathematical Society. In addition, Lang is recognized as a Hinge Visible Expert, as an engineer who has attained high visibility and expertise; the Complete Book of Origami. A K Peters. 2003. ISBN 1-56881-194-2. Napkin folding problem Official website Doctoral Thesis Abstract Computational Origami from IT Conversations Robert Lang at TED Radio interview at The Connection Interview with Robert Lang, by Margaret Wertheim, featured in Cabinet magazine, Issue 17, Spring 2005 Origami Engineering in the Fold: video report on origami telescopes An Origami Space Telescope Interview with Peter Shea at Institute for Advanced Study, University of Minnesota, March 2011 Origami^6, American Math Society, Origami Insects II.
Joseph-Louis Lagrange was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, both classical and celestial mechanics. In 1766, on the recommendation of Leonhard Euler and d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics, written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century. In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences, he remained in France until the end of his life. He was involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, became Senator in 1799.
Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations, he proved. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series, he studied the three-body problem for the Earth and Moon and the movement of Jupiter's satellites, in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. But above all, he is best known for his work on mechanics, where he transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, presented the so-called mechanical "principles" as simple results of the variational calculus.
Born as Giuseppe Lodovico Lagrangia, Lagrange was of French descent. His paternal great-grandfather was a French army officer who had moved to Turin, the de facto capital of the kingdom of Piedmont-Sardinia at Lagrange's time, married an Italian, his mother was from the countryside of Turin. He was raised as a Roman Catholic, his father, who had charge of the king's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained a good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father, Lagrange seems to have accepted this willingly, he studied at the University of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, it was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident.
Alone and unaided he threw himself into mathematical studies. Charles Emmanuel III appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler. In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni, the academy's military commander and famous artillery theorist, Lagrange proved to be a problematic professor with his oblivious teaching style, abstract reasoning, impatience with artillery and fortification-engineering applications. In this Academy one of his students was François Daviet de Foncenex. Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, discovering a method of maximising and minimising functionals in a way similar to finding extrema of functions.
Lagrange wrote several letters to Leonhard Euler between 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and simplifying Euler's earlier analysis. Lagrange applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis. Euler was impressed with Lagrange's results, it has been stated that "with characteristic courtesy he withheld a paper he had written, which covered some of the same ground, in order that the young Italian might have time to complete his work, claim the undisputed invention of the new calculus". Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773. In 1758, with the aid of his pupils, Lagrange established a society, subsequently incorporated as the Turin Aca
In geometry, the Tammes problem is a problem in packing a given number of circles on the surface of a sphere such that the minimum distance between circles is maximized. It is named after a Dutch botanist who posed the problem in 1930 while studying the distribution of pores on pollen grains, it can be viewed as a particular special case of the generalized Thomson problem. Spherical code Kissing number problem Journal articlesTammes PML. "On the origin of number and arrangement of the places of exit on pollen grains". Diss. Groningen. Tarnai T. "Multi-symmetric close packings of equal spheres on the spherical surface". Acta Crystallographica. A43: 612–616. Doi:10.1107/S0108767387098842. Erber T, Hockney GM. "Equilibrium configurations of N equal charges on a sphere". Journal of Physics A: Mathematical and General. 24: Ll369–Ll377. Bibcode:1991JPhA...24L1369E. Doi:10.1088/0305-4470/24/23/008. Melissen JBM. "How Different Can Colours Be? Maximum Separation of Points on a Spherical Octant". Proceedings of the Royal Society A. 454: 1499–1508.
Bibcode:1998RSPSA.454.1499M. Doi:10.1098/rspa.1998.0218. Bruinsma RF, Gelbart WM, Reguera D, Rudnick J, Zandi R. "Viral Self-Assembly as a Thermodynamic Process". Physical Review Letters. 90: 248101–1–248101–4. ArXiv:cond-mat/0211390. Bibcode:2003PhRvL..90x8101B. Doi:10.1103/PhysRevLett.90.248101. Archived from the original on 2007-09-15. BooksAste T, Weaire DL; the Pursuit of Perfect Packing. Taylor and Francis. Pp. 108–110. ISBN 978-0-7503-0648-5. Wells D; the Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. P. 31. ISBN 0-14-011813-6. How to Stay Away from Each Other in a Spherical Universe. Packing and Covering of Congruent Spherical Caps on a Sphere. Talk on the Tammes problem. Science of Spherical Arrangements. General discussion of packing points on surfaces, with focus on tori