Mehmet Koca - Academia.edu (original) (raw)

Papers by Mehmet Koca

Research paper thumbnail of Noncrystallographic Coxeter group H4 in E8

Journal of Physics A-mathematical and General, 2001

The E8 lattice is constructed in terms of icosians by matching two sets of F4 lattices described ... more The E8 lattice is constructed in terms of icosians by matching two sets of F4 lattices described by quaternions. Embedding the noncrystallographic group H4 into the Weyl group W(E8) has been described using matrix generators with an emphasis on the relevant Coxeter elements. The conjugacy classes of H4 in terms of quaternions and the characters of the two four-dimensional irreducible representations are explicitly calculated.

Research paper thumbnail of The Chevalley group G 2(2) of order 12096 and the octonionic root system of E 7

Linear Algebra and Its Applications, 2007

The octonionic root system of the exceptional Lie algebra E8 has been constructed from the quater... more The octonionic root system of the exceptional Lie algebra E8 has been constructed from the quaternionic roots of F4 using the Cayley-Dickson doubling procedure where the roots of E7 correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E7 is the adjoint Chevalley group G2(2) of order 12096. One of the four maximal subgroups of G2(2) of order 192 preserves the quaternion subalgebra of the E7 root system. The other three maximal subgroups of orders 432, 192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E6 × U (1), SU (2) × SO(12) and SU (8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension.

Research paper thumbnail of Quasi regular polygons and their duals with Coxeter symmetries Dn represented by complex numbers

Journal of Physics: Conference Series, 2011

In two series of papers we construct quasi regular polyhedra and their duals which are similar to... more In two series of papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of graphene where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagrams.

Research paper thumbnail of Quaternionic Roots of E 8 Related Coxeter Graphs and Quasicrystals

The lattice matching of two sets of quaternionic roots of F4 leads to quaternionic roots of E8 wh... more The lattice matching of two sets of quaternionic roots of F4 leads to quaternionic roots of E8 which has a decomposition H4 + σH4 where the Coxeter graph H4 is represented by the 120 quaternionic elements of the binary icosahedral group. The 30 pure imaginary quaternions constitute the roots of H3 which has a natural extension to H3 + σH3 describing the root system of the Lie algebra D6 . It is noted that there exist three lattices in 6-dimensions whose point group W (D6) admits the icosahedral symmetry H3 as a subgroup, the roots of which describe the mid-points of the edges of an icosahedron. A natural extension of the Coxeter group H2 of order 10 is the Weyl group W (A4) where H2 + σH2 constitute the root system of the Lie algebra A4 . The relevance of these systems to quasicrystals are discussed.

Research paper thumbnail of Grand antiprism and quaternions

Journal of Physics A-mathematical and General, 2009

Vertices of the four-dimensional (4D) semi-regular polytope, the grand antiprism and its symmetry... more Vertices of the four-dimensional (4D) semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E8 root system which decomposes into two copies of the root system of H4. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W(H4). It is the group Aut(H2 ⊕ H'2) which is constructed in terms of 20 quaternionic roots of the Coxeter diagram H2 ⊕ H'2. The root system of H4 represented by the binary icosahedral group I of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram H2 ⊕ H'2 are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has also been constructed.

Research paper thumbnail of Maximal subgroups of the Coxeter group W( H 4) and quaternions

Linear Algebra and Its Applications, 2006

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W (H 4 ) of order 1... more The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W (H 4 ) of order 14,400. Its derived subgroup is the largest finite subgroup W (H 4 )/Z 2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W (H 2 ) × W (H 2 )] Z 4 and W (H 3 ) × Z 2 possess noncrystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU (3) × SU (3), SU (5) and SO(8) respectively. We represent the maximal subgroups of W (H 4 ) with sets of quaternion pairs acting on the quaternionic root systems.

Research paper thumbnail of Maximal subgroups of the Coxeter group () and quaternions

Linear Algebra and Its Applications, 2006

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W (H 4 ) of order 1... more The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W (H 4 ) of order 14,400. Its derived subgroup is the largest finite subgroup W (H 4 )/Z 2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W (H 2 ) × W (H 2 )] Z 4 and W (H 3 ) × Z 2 possess noncrystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU (3) × SU (3), SU (5) and SO(8) respectively. We represent the maximal subgroups of W (H 4 ) with sets of quaternion pairs acting on the quaternionic root systems.

Research paper thumbnail of Changes in the engineering properties of marble in fire-exposed columns

International Journal of Rock Mechanics and Mining Sciences, 2006

Two marble columns in the bottom floor of a school building were exposed to intensive heat in the... more Two marble columns in the bottom floor of a school building were exposed to intensive heat in the fire, and were subject of a debate about their load-carrying capacity following the event. After their removal from their original spots, series of tests on column samples were carried out to investigate variations in their engineering characteristics due to fire hazard. The temperature at the surface of the marble columns hit by the flames directly during the fire was estimated to be on the order of 500 1C using an empirically established relationship between porosity and temperature on intact marble core samples. Results of mechanical tests indicated that marble columns were still holding a significant load-carrying capacity despite 23% loss of strength at fire temperature. The spallings at the corners of the blocks forced the board of education to give the decision of replacing the gray marble columns with steel members. Occurrence of these spallings is attributed to stress concentrations and widening of micro fractures. Based on the results of mechanical, physical and chemical tests, a methodology to assess influence of fire hazard on marble and similar geological materials was developed. r

Research paper thumbnail of Engineering-geological and geotechnical investigations for risk assessment of the University Olympic Village in Izmir (Turkey

Geologos, 2010

The Metropolitan Municipality of Izmir (Turkey) designated a steeply inclined area for the constr... more The Metropolitan Municipality of Izmir (Turkey) designated a steeply inclined area for the construction of buildings to house the participants of the Izmir University Olympic Games. Before the construction activities could start, engineering geological and geotechnical investigations had to be carried out in order to establish which zones in the area were suitable for safe constructions. Seismic studies, borings and laboratory tests yielded the data, which were used for preparing five hazard maps in a GIS environment. The construction activities based on the results of this complex investigation appeared successful. The engineering geological investigations included geotechnical measurements on core samples obtained from the boreholes (core drilling) and laboratory testing. The rock-quality designation (RQD%) values of the rock units were determined and used in the rock-mass classification (rock-mass rating method) as an input parameter and in the calculation of the bearing capacity of the various rock units. Geophysical surveys were carried out to determine the seismic velocity of the rocks at the site. A short overview is provided of the main problems that had to be dealt with, and of the successive steps taken to solve the engineering-geological problems. Determination of these problems is necessary for adequate land-use planning and construction activities.

Research paper thumbnail of Large-scale land-suitability mapping in the GIS environment for the construction site of the University Olympic Village in Izmir (Turkey

Geologos, 2009

Participants of the 23 rd University Olympic Games held in Izmir (Turkey) were hosted in a Univer... more Participants of the 23 rd University Olympic Games held in Izmir (Turkey) were hosted in a Universiade Olympic Village that had to be newly constructed. A risk analysis has been performed to assess potential engineering problems at the envisaged construction site. The risk analysis was largely based on a new, large-scale land-suitability map that was prepared in the GIS environment with much detail for this type of construction work. Information was collected from several maps, in combination with mapping in the field, drilling and seismic data. Geological, geotechnical, geophysical and morphological data were then superimposed on these analyses. This sequence of overlay analyses was performed with the help of GIS software (MapInfo Professional 7.5); this resulted in five hazard maps. Risk points (1-11) were then attributed to the different zones in the five digital hazard maps. A land-suitability map indicating the suitability for envisaged constructional activities was subsequently obtained in the form of an overlay of the five hazard maps, thus allowing to calculate a total risk for each zone on this map. The land-suitability map that was thus obtained, has been prepared for a 1:1,000 scale development plan; such a large scale is uncommon in this context.

Research paper thumbnail of Anchor application in Karatepe andesite rock slope, Izmir—Türkiye

International Journal of Rock Mechanics and Mining Sciences, 2011

Atatürk's Monument is a steel-constructed building containing a space-truss system, and is t... more Atatürk's Monument is a steel-constructed building containing a space-truss system, and is the tenth highest monumental construction in the world. Atatürk's mask is installed on this construction. The monument was planned to be built on rocky ground in the old quarry ...

Research paper thumbnail of Noncrystallographic Coxeter group H4 in E8

Journal of Physics A-mathematical and General, 2001

The E8 lattice is constructed in terms of icosians by matching two sets of F4 lattices described ... more The E8 lattice is constructed in terms of icosians by matching two sets of F4 lattices described by quaternions. Embedding the noncrystallographic group H4 into the Weyl group W(E8) has been described using matrix generators with an emphasis on the relevant Coxeter elements. The conjugacy classes of H4 in terms of quaternions and the characters of the two four-dimensional irreducible representations are explicitly calculated.

Research paper thumbnail of The Chevalley group G 2(2) of order 12096 and the octonionic root system of E 7

Linear Algebra and Its Applications, 2007

The octonionic root system of the exceptional Lie algebra E8 has been constructed from the quater... more The octonionic root system of the exceptional Lie algebra E8 has been constructed from the quaternionic roots of F4 using the Cayley-Dickson doubling procedure where the roots of E7 correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E7 is the adjoint Chevalley group G2(2) of order 12096. One of the four maximal subgroups of G2(2) of order 192 preserves the quaternion subalgebra of the E7 root system. The other three maximal subgroups of orders 432, 192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E6 × U (1), SU (2) × SO(12) and SU (8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension.

Research paper thumbnail of Quasi regular polygons and their duals with Coxeter symmetries Dn represented by complex numbers

Journal of Physics: Conference Series, 2011

In two series of papers we construct quasi regular polyhedra and their duals which are similar to... more In two series of papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of graphene where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagrams.

Research paper thumbnail of Quaternionic Roots of E 8 Related Coxeter Graphs and Quasicrystals

The lattice matching of two sets of quaternionic roots of F4 leads to quaternionic roots of E8 wh... more The lattice matching of two sets of quaternionic roots of F4 leads to quaternionic roots of E8 which has a decomposition H4 + σH4 where the Coxeter graph H4 is represented by the 120 quaternionic elements of the binary icosahedral group. The 30 pure imaginary quaternions constitute the roots of H3 which has a natural extension to H3 + σH3 describing the root system of the Lie algebra D6 . It is noted that there exist three lattices in 6-dimensions whose point group W (D6) admits the icosahedral symmetry H3 as a subgroup, the roots of which describe the mid-points of the edges of an icosahedron. A natural extension of the Coxeter group H2 of order 10 is the Weyl group W (A4) where H2 + σH2 constitute the root system of the Lie algebra A4 . The relevance of these systems to quasicrystals are discussed.

Research paper thumbnail of Grand antiprism and quaternions

Journal of Physics A-mathematical and General, 2009

Vertices of the four-dimensional (4D) semi-regular polytope, the grand antiprism and its symmetry... more Vertices of the four-dimensional (4D) semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E8 root system which decomposes into two copies of the root system of H4. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W(H4). It is the group Aut(H2 ⊕ H'2) which is constructed in terms of 20 quaternionic roots of the Coxeter diagram H2 ⊕ H'2. The root system of H4 represented by the binary icosahedral group I of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram H2 ⊕ H'2 are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has also been constructed.

Research paper thumbnail of Maximal subgroups of the Coxeter group W( H 4) and quaternions

Linear Algebra and Its Applications, 2006

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W (H 4 ) of order 1... more The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W (H 4 ) of order 14,400. Its derived subgroup is the largest finite subgroup W (H 4 )/Z 2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W (H 2 ) × W (H 2 )] Z 4 and W (H 3 ) × Z 2 possess noncrystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU (3) × SU (3), SU (5) and SO(8) respectively. We represent the maximal subgroups of W (H 4 ) with sets of quaternion pairs acting on the quaternionic root systems.

Research paper thumbnail of Maximal subgroups of the Coxeter group () and quaternions

Linear Algebra and Its Applications, 2006

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W (H 4 ) of order 1... more The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W (H 4 ) of order 14,400. Its derived subgroup is the largest finite subgroup W (H 4 )/Z 2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W (H 2 ) × W (H 2 )] Z 4 and W (H 3 ) × Z 2 possess noncrystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU (3) × SU (3), SU (5) and SO(8) respectively. We represent the maximal subgroups of W (H 4 ) with sets of quaternion pairs acting on the quaternionic root systems.

Research paper thumbnail of Changes in the engineering properties of marble in fire-exposed columns

International Journal of Rock Mechanics and Mining Sciences, 2006

Two marble columns in the bottom floor of a school building were exposed to intensive heat in the... more Two marble columns in the bottom floor of a school building were exposed to intensive heat in the fire, and were subject of a debate about their load-carrying capacity following the event. After their removal from their original spots, series of tests on column samples were carried out to investigate variations in their engineering characteristics due to fire hazard. The temperature at the surface of the marble columns hit by the flames directly during the fire was estimated to be on the order of 500 1C using an empirically established relationship between porosity and temperature on intact marble core samples. Results of mechanical tests indicated that marble columns were still holding a significant load-carrying capacity despite 23% loss of strength at fire temperature. The spallings at the corners of the blocks forced the board of education to give the decision of replacing the gray marble columns with steel members. Occurrence of these spallings is attributed to stress concentrations and widening of micro fractures. Based on the results of mechanical, physical and chemical tests, a methodology to assess influence of fire hazard on marble and similar geological materials was developed. r

Research paper thumbnail of Engineering-geological and geotechnical investigations for risk assessment of the University Olympic Village in Izmir (Turkey

Geologos, 2010

The Metropolitan Municipality of Izmir (Turkey) designated a steeply inclined area for the constr... more The Metropolitan Municipality of Izmir (Turkey) designated a steeply inclined area for the construction of buildings to house the participants of the Izmir University Olympic Games. Before the construction activities could start, engineering geological and geotechnical investigations had to be carried out in order to establish which zones in the area were suitable for safe constructions. Seismic studies, borings and laboratory tests yielded the data, which were used for preparing five hazard maps in a GIS environment. The construction activities based on the results of this complex investigation appeared successful. The engineering geological investigations included geotechnical measurements on core samples obtained from the boreholes (core drilling) and laboratory testing. The rock-quality designation (RQD%) values of the rock units were determined and used in the rock-mass classification (rock-mass rating method) as an input parameter and in the calculation of the bearing capacity of the various rock units. Geophysical surveys were carried out to determine the seismic velocity of the rocks at the site. A short overview is provided of the main problems that had to be dealt with, and of the successive steps taken to solve the engineering-geological problems. Determination of these problems is necessary for adequate land-use planning and construction activities.

Research paper thumbnail of Large-scale land-suitability mapping in the GIS environment for the construction site of the University Olympic Village in Izmir (Turkey

Geologos, 2009

Participants of the 23 rd University Olympic Games held in Izmir (Turkey) were hosted in a Univer... more Participants of the 23 rd University Olympic Games held in Izmir (Turkey) were hosted in a Universiade Olympic Village that had to be newly constructed. A risk analysis has been performed to assess potential engineering problems at the envisaged construction site. The risk analysis was largely based on a new, large-scale land-suitability map that was prepared in the GIS environment with much detail for this type of construction work. Information was collected from several maps, in combination with mapping in the field, drilling and seismic data. Geological, geotechnical, geophysical and morphological data were then superimposed on these analyses. This sequence of overlay analyses was performed with the help of GIS software (MapInfo Professional 7.5); this resulted in five hazard maps. Risk points (1-11) were then attributed to the different zones in the five digital hazard maps. A land-suitability map indicating the suitability for envisaged constructional activities was subsequently obtained in the form of an overlay of the five hazard maps, thus allowing to calculate a total risk for each zone on this map. The land-suitability map that was thus obtained, has been prepared for a 1:1,000 scale development plan; such a large scale is uncommon in this context.

Research paper thumbnail of Anchor application in Karatepe andesite rock slope, Izmir—Türkiye

International Journal of Rock Mechanics and Mining Sciences, 2011

Atatürk's Monument is a steel-constructed building containing a space-truss system, and is t... more Atatürk's Monument is a steel-constructed building containing a space-truss system, and is the tenth highest monumental construction in the world. Atatürk's mask is installed on this construction. The monument was planned to be built on rocky ground in the old quarry ...