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### There are integral heptagons, no three points on a line, no four on a circle [article]

Tobias Kreisel, Sascha Kurz
2008 arXiv   pre-print
We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erdős.  ...  So now we ask as a substitute: " Are there eight points in the plane, no three on a line, no four on a circle with pairwise integral distances?"  ...  Up to diameter 20000 there are only few integral point sets consisting of 6 points, no three on a line, no four on a circle with pairwise integral distances, see  for a complete list.  ...

### There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle

Tobias Kreisel, Sascha Kurz
2007 Discrete & Computational Geometry
We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erdős.  ...  So now we ask as a substitute: "Are there eight points in the plane, no three on a line, no four on a circle with pairwise integral distances?"  ...  Introduction A famous open problem of P. Erdős asks for seven points in the plane, no three on a line, no four on a circle with pairwise rational or integral distances [1, 3] .  ...

### Encounter at far point

Michael Kleber
2008 The Mathematical intelligencer
We are now picky enough to want sets of points with pairwise rational distances and with no three on a line, no four on a circle -that is, "in general position," the way some people use the phrase.  ...  Well, infinitely many along any line, of course, so we require sets of points with no three collinear. It turns out to be easy if you allow points that all lie on a circle, too.  ...  We are now picky enough to want sets of points with pairwise rational distances and with no three on a line, no four on a circle -that is, "in general position," the way some people use the phrase.  ...

### From Pentagon to Heptagon: A Discovery on the Generation of the Regular Heptagon from the Equilateral Triangle and Pentagon

Mark A. Reynolds
2001 Nexus Network journal
There are a variety of procedures, but I find this one to be of interest because of its utilization of the vesica piscis and the circle.  ...  With four of the seven sides of the heptagon completed by the two bisections, and using the lengths of each of the four completed sides of the heptagon to obtain the final three sides by measuring them  ...  There is no question of approximation or an issue of speculation. This is an ideal situation. We have φ in both its major roles: mathematical and organic.  ...

### Constructing random polygons

David Dailey, Deborah Whitfield
2008 Proceedings of the 9th ACM SIGITE conference on Information technology education - SIGITE '08
The method and its applications are simple, easy to manipulate and requires no construction of angles except 90 o for the construction of the nuclear figure the square.  ...  The method uses the diagonal of the square to determine the radius of a circle that is equally sectored using the side of the square.  ...  There is no ready explanation to any of these discrepancies or observations. The addition of a quarter of OB to point A down ward, figure 3 , gave point W.  ...

### Conformal mapping of rectangular heptagons

Andrei B Bogatyrev
2012 Sbornik. Mathematics
We propose an analytical approach to the conformal mapping of (rectangular) polygons based on the theory of Riemann surfaces and theta functions.  ...  Moreover, this polygonal line has no selfintersections.  ...  However there is a single valued branch of the integral in the disc H + which maps the latter 1 − 1 to the given normalized heptagon.  ...

### Elementary Problems and Solutions

1957 The American mathematical monthly
Then, if there is only one fixed point of G, the centroids of the two sets of points 4, +--+ ,c,and d,---,d, coincide.  ...  Prove that the difference between two positive integral powers of the same integer is exactly divisible by six unless the integer gives the remainder two on division by three and one power is odd while  ...

### Pleats in crystals on curved surfaces

William T. M. Irvine, Vincenzo Vitelli, Paul M. Chaikin
2010 Nature
In a hexagonal lattice in which every particle has six nearest neighbours (Fig. 1, inset) , there are two types of topological defects (Fig. 2) : disclinations that disrupt orientational order and appear  ...  as points of local five-fold or seven-fold symmetry, (pentagons or heptagons, having topological charge 6(2p/6), and dislocations, which disrupt translational order and appear as disclination dipoles  ...  That is, for a full sphere there are 12 pentagons (1p/3)s, for a hemisphere there are 6 pentagons (1p/3)s and on smaller fractions the two remain approximately proportional.  ...

### A Classification Scheme for Toroidal Molecules [article]

Jorge Berger, Joseph E. Avron
1995 arXiv   pre-print
Each tiling is characterized by a set of four integers and determines a toroidal molecule.  ...  We construct a class of periodic tilings of the plane, which corresponds to toroidal arrangements of trivalent atoms, with pentagonal, hexagonal and heptagonal rings.  ...  Through every point P on the surface of the torus there is a circle C 2 of radius R/η in a plane that contains the z axis; the center of C 2 belongs to C 1 . C 2 is a longitudinal line.  ...

### Geometry, Light, and Cosmology in the Church of Hagia Sophia

Wassim Jabi, Iakovos Potamianos
2007 International Journal of Architectural Computing
While we have no written proof that Anthemius used this reflector for Hagia Sophia, the concern with the seasons and shining a light on a specific point is consistent with a concern for the liturgical  ...  "  Our research points to these three concerns as the major factors in designing the church.  ...

### ON THE PENTAGRAM AS A PYTHAGOREAN EMBLEM

John A. Fossa
2020 Revista brasileira de história da matemática (RBHM)
Uma investigação das propriedades matemáticas do pentágono mostra que ele foi usado como um emblema pitagórica para representar a metempsicose.  ...  There are so many diagonals in the heptagon that it is not readily clear what is happening in Figure 3 .  ...  In fact, however, the heptagon's diagonals reproduce the heptagon twiceonce by a starred heptagon with long arms and once by one with short arms (inside a seven pointed star with very long arms).  ...

### Role of lattice defects in catalytic activities of graphene clusters for fuel cells

Lipeng Zhang, Quan Xu, Jianbing Niu, Zhenhai Xia
2015 Physical Chemistry, Chemical Physics - PCCP
Defects are common but important in graphene, which could significantly tailor the electronic structures and physical and chemical properties.  ...  There are no energy barriers found on PZ and GLD-558-01 graphene clusters in the third electron transfer step.  ...  The four-electrons and two- (e) shows a pentagon ring at a zigzag edge. One-dimensional defects, also called line defects, can be thought of as a line of reconstructed point defects.  ...

### Phyllotaxis on surfaces of constant Gaussian curvature

J F Sadoc, J Charvolin, N Rivier
2013 Journal of Physics A: Mathematical and Theoretical
A close packed organization with circular symmetry of a large number of small discs on a plane is obtained when the centres of the discs are distributed according to the algorithm of phyllotaxis.  ...  The evolutions of the local order within the grains show no differences except on the equatorial belt of the sphere where the isotropy is decreased owing to the symmetry of this finite surface around its  ...  Blue, red and green cells are respectively pentagons, hexagons and heptagons. The three spirals joining the first neighbours of the points are called parastichies.  ...

### Frequency tunable topological edge states of two-dimensional honeycomb lattice photonic crystals [article]

Yuchen Peng, Bei Yan, Jianlan Xie, Exian Liu, Hang Li, Rui Ge, Feng Gao, Jianjun Liu
2019 arXiv   pre-print
The topological edge states and their unidirectional transmission as the scatterers with triangular, pentagonal, and heptagonal shapes are researched.  ...  Combining two honeycomb lattice PCs with a common photonic band gap (PBG) but different band topologies can generate a topologically protected edge state at the combined junction.  ...  Later, there are a series of research on the PQSHE using the honeycomb lattice model, such as, using air hole scatterers    , triangular scatterers [24, 26] , realizing asymmetric radiation  ...

### Advanced Problems and Solutions

1957 The American mathematical monthly
If 2;;, denotes the Kantor point of a triangle A;A;A, with respect to the quadrangle formed by the remaining four vertices, then the thirty-five points 0;; all lie on a circle one-half as large as the  ...  prove that the ten points Q,; all lie on a circle.  ...
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