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

2008
*
arXiv
*
pre-print

We give two configurations of seven

arXiv:0804.1303v1
fatcat:6niexf7glrcb5hhpcwv574mwfy
*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 [8] for*a*complete list. ...##
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There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle

2007
*
Discrete & Computational Geometry
*

We give two configurations of seven

doi:10.1007/s00454-007-9038-6
fatcat:3mxxyhowdjb27pxgrr4dkxqh7m
*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] . ...##
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Encounter at far point

2008
*
The Mathematical intelligencer
*

We

doi:10.1007/bf02985756
fatcat:j2t2hukdlzh2bn6spg5mglruke
*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. ...##
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From Pentagon to Heptagon: A Discovery on the Generation of the Regular Heptagon from the Equilateral Triangle and Pentagon

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. ...

##
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Constructing random polygons

2008
*
Proceedings of the 9th ACM SIGITE conference on Information technology education - SIGITE '08
*

The method and its applications

doi:10.1145/1414558.1414592
dblp:conf/sigite/DaileyW08
fatcat:jo7gdaw5mrh5rpfhlyj5viua4q
*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. ...##
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Conformal mapping of rectangular heptagons

2012
*
Sbornik. Mathematics
*

We propose an analytical approach to the conformal mapping of (rectangular) polygons based

doi:10.1070/sm2012v203n12abeh004284
fatcat:7w5dd2y5dnd65ke6blhbrk544m
*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*. ...##
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Elementary Problems and Solutions

1957
*
The American mathematical monthly
*

Then, if

doi:10.1080/00029890.1957.11988942
fatcat:rck5paofdfaehjfgpc6kztolsi
*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 ...##
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Pleats in crystals on curved surfaces

2010
*
Nature
*

In

doi:10.1038/nature09620
pmid:21164482
fatcat:yn4g7oginjetjcho4fgmuboz74
*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. ...##
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A Classification Scheme for Toroidal Molecules
[article]

1995
*
arXiv
*
pre-print

Each tiling is characterized by

arXiv:mtrl-th/9502001v2
fatcat:hap6cvms2bepdg4oycejtmyz24
*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*. ...##
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Geometry, Light, and Cosmology in the Church of Hagia Sophia

2007
*
International Journal of Architectural Computing
*

While we have

doi:10.1260/1478-0771.5.2.304
fatcat:3lpojflu3fgttdt2gvkwkxzorm
*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 ... " [4] Our research*points*to these*three*concerns as the major factors in designing the church. ...##
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ON THE PENTAGRAM AS A PYTHAGOREAN EMBLEM

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

doi:10.47976/rbhm2006v6n12127-137
doaj:35cb2e7bf15745f99335876c8cdeda2e
fatcat:6jugzrtu3za65gbuufnz6tq6g4
*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). ...##
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Role of lattice defects in catalytic activities of graphene clusters for fuel cells

2015
*
Physical Chemistry, Chemical Physics - PCCP
*

Defects

doi:10.1039/c5cp02014j
pmid:26033301
fatcat:bstags3mf5gxpjs7atfsvqtovy
*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. ...##
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Phyllotaxis on surfaces of constant Gaussian curvature

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. ...

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

2019
*
arXiv
*
pre-print

The topological edge states and their unidirectional transmission as the scatterers with triangular, pentagonal, and

arXiv:1912.08078v1
fatcat:h4ugpmwzhbcz7kwmbvcqnwg4nq
*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 [24] [25] [26] , triangular scatterers [24, 26] , realizing asymmetric radiation ...##
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Advanced Problems and Solutions

1957
*
The American mathematical monthly
*

If 2;;, denotes the Kantor

doi:10.1080/00029890.1957.11988960
fatcat:r4m66shevrfctf3ce63gg4i5fy
*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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