Geometric Constitution of Space Structure Based on Regular-Polyhedron Combinations

Transcription

1 Geometric Constitution of Spce Structure Bsed on Regulr-Polyhedron Combintions Zichen Wng School of Civil Engineering nd Trnsporttion South Chin University of Technology Gungzhou, Gungdong, Chin Abstrct In order to chieve unique rchitecturl effect, the design dopts the semless combintions, series of rrys, rottions nd the cutting of truncted octhedrons, cubocthedrons nd truncted tetrhedrons before the spce plte structures or frme structures re generted. By chnging the rottion xis, the rottion ngle nd the position of the cutting plnes of polyhedrl combined structures, we hve obtined the structures nd the cutting surfce ptterns which stisfy prcticl engineering significnce. Studies hve shown tht verticl to coordinte xes, perpendiculr to body digonl, rotting round coordinte xes nd revolving round the (,,) to (,,) vector xis, the four cutting modes ll get spce structures nd fncy cutting surfce ptterns which cn meet the engineering needs, providing chrcteristic smples for building elevtions nd roof ptterns. Keywords - truncted octhedron, cubocthedron, truncted tetrhedron, geometric constitution, cutting surfce ptterns I. INTRODUCTION In 9, t the Second Interntionl Congress of Mthemticins, D. Hilbert put forwrd the fmous questions which influence the whole th century s mthemticl development. Of these questions, the eighteenth question is to construct spce with the sme polyhedrons while Kelvin, n Irish mthemticin, put forwrd conjecture: Remove rectngulr pyrmid whose side length is one-third of the side length of the regulr octhedron on the six vertices of the regulr octhedron, nd we will get tetrkidechedron, nd this ws regrded s the best nswer[] until yers lter in 99, when D. Weire nd R. Pheln, two Americn mthemticins, gve counter-exmple to the Kelvin conjecture[]. They dopted combintion of tetrkidechedrons (this differs from Kelvin's tetrkidechedron) with regulr dodechedron, or clled the Weire-pheln polyhedron (WP polyhedron),nd the contct re is pproximtely.% less thn tht of the Kelvin s structure nd this is the best nswer to the Kelvin question t present. On the other hnd, these resonble spce structure forms cn lso be esily found in nture. For exmple, combustible ice (lso known s frozen methne nd wter) is n ccumultion of WP polyhedrons. Alloys like Cs 8 Sn, K 8 Si, nd N 8 Si re lso consistent with the collective structures of the WP polyhedron; the truncted octhedron nd the crystlline compound Mg (Zn, Al) shre the sme structure []; the sodlite is semlessly structured by truncted octhedron, while the beehives nd the grnet contin rhombic dodechedron structures []. Undoubtedly, ny structure s survivl must hve experienced long-term nturl selection, so it is of gret significnce to pply these structures to humn rchitecturl science. Beijing s fmous Wter Cube, or Chin s Ntionl Swimming Center, hs tken the dvntge of the WP polyhedron technology. The ctul geometry is creting n rry of optimized Weire Pheln fom which is much bigger thn the building proper, rotting the rry bout n xis, nd then trimming the fom tht flls either outside the building envelope or within the occupied spces. The resulting structure forms the roof nd wlls of the building. By optimizing the formtion to form qusi WP polyhedron, fmous building, its first kind in Chin, cme into being fter n effective cutting of pile of polyhedrons. Zhejing University s Scholr Xing Xi n et l. hs studied in gret detil spce frme structures formed by two filling solids of the truncted octhedron nd rhombic dodechedron []. If the pckings of the truncted octhedron nd rhombic dodechedron re rotted round the (,, ) to (,, ) vector xis with n ngle of nd then cut through the pproprite plnes, new polyhedrl spce frmes which hve repetitive structures nd meet construction requirements cn be obtined to show rndom nd complex pttern on surfces. As the reserch on spce structures formed by the strong symmetry of polyhedron combintions is scrce both t home nd brod, this present pper, bsed on the reserch of Pltonic solids nd Archimeden solids, is intended to mke systemtic nlysis of the geometry of the spce structure formed by the combintion of three kinds of regulr polyhedrons (coming from Pltonic solids nd Archimeden solids). Menwhile, we get mny kinds of spce structures, which re pplicble to the prcticl needs of mny kinds of engineering tht require novel ptterns on their cutting surfces(for exmple, like the ptterns of broken glss). As result, the study helps to enrich the reserch of the spce structures of polyhedrons. DOI./IJSSST.... ISSN: -8x online, -8 print

2 II. SPACE FILLING BETWEEN PLATONIC SOLIDS AND ARCHIMEDEAN SOLIDS Pltonic solids nd Archimeden solids re regulr polyhedrons, for they re highly symmetricl nd the polyhedrl ngles t ech vertex re exctly the sme. Their dihedrl ngles cn be clculted s shown in the following tbles. (Angle TABLE I. TABLE II. Nomenclture Truncted tetrhedron Cubocthedron Truncted octhedron Truncted cube THE DIHEDRAL ANGLE OF PLATONIC SOLIDS Nomenclture The Cosine of Dihedrl Angle nd Its Vlue Regulr cos, i.e.,. tetrhedron Regulr hexhedron 9 Regulr cos, i.e., 9. octhedron Regulr dodechedron cos, i.e.,. Regulr icoshedrons cos, i.e., 8.9 THE DIHEDRAL ANGLE OF ARCHIMEDEAN SOLIDS The Cosine of Dihedrl Angle nd Its Vlue cos,. cos, 9. cos,. cos,. cos, 9. cos 88, 88 9 cos 8, 8. Rhombicuboct cos,. ; hedron Truncted cubocthedron cos 8, 8. cos 8, 8 cos,. mn represents dihedrl ngle of regulr polygon of m sides nd regulr polygon of n sides) According to the dihedrl ngles on the tbles bove, we cn see tht there exist kinds of semless spce filling combintion models between Pltonic solids nd Archimeden solids. These combintions re s follows: () the combintion of regulr tetrhedron nd regulr octhedron; () the combintion of regulr tetrhedron nd truncted tetrhedron; ()the combintion of cubocthedron nd regulr octhedron; ()the combintion of truncted cube nd regulr octhedron; ()the combintion of truncted tetrhedron, cubocthedron nd truncted octhedron; ()the combintion of rhombicubocthedron, cube nd cubocthedron; () the combintion of rhombicubocthedron, cube nd regulr tetrhedron; (8) the combintion of rhombicubocthedron, truncted cube, cube nd regulr octgonl prism; (9) the combintion of truncted cubocthedron, truncted cube nd truncted tetrhedron; ()the combintion of truncted cubocthedron, cube nd truncted octhedron; () the combintion of Truncted cubocthedron nd regulr octgonl prism. III. SPACE STRUCTURES BASED ON THE COMBINATION OF TRUNCATED TETRAHEDRON, CUBOCTAHEDRON AND TRUNCATED OCTAHEDRON The spce structures of polyhedrons, studied in this pper, include the plte structures nd the frme structures. When cutting polyhedrons in frme structure, we cn get the surfce chords mde up of the intersecting lines of cutting surfce nd bdominl rods mde up of the originl edges keeping in interior of polyhedrons []. The spce structures of polyhedrons re complex nd complicted. In order to estblish resonble structure which cn provide theoreticl bsis nd fesible method to meet the rchitecturl engineering effect, nd enrich ll kinds of new spce structures of polyhedrons, this pper minly studies the following spects: () Mke the bsic units nd combintoril nlysis of spce structures of polyhedrons, the distribution nd rry of combintions of bsic elements in spce structures; () Explore vrious cutting surfces of spce structures of polyhedrons, in order to look for the cutting surfce ptterns (roofing or metope), which meets rchitecturl requirements; () Explore the reltionships between concrete building mesurements nd the size of the Geometry of structure, nd the minimum distnce between n internl node nd the cutting surfce, etc. A. Bsic Unit Anlysis nd Anlysis of Spce Polyhedrons Combintion On the four vertices of regulr tetrhedron, remove regulr tetrhedron whose edge length is one-third edge length of the regulr tetrhedron, nd we will get truncted tetrhedron. There re four regulr tringulr plnes nd four regulr hexgon plnes, nd the distnce between two prllel plnes is times of the edge length. The cubocthedron is got by cutting regulr tringulr pyrmid whose length is hlf of the cube t the eight vertexes of the cube, nd the positive view, top view nd side view re the sme. The squre plne is in prllel to its corresponding squre plne, while the distnce between the two prllel plnes is times of the edge length; the DOI./IJSSST.... ISSN: -8x online, -8 print

3 tringle plne is in prllel to its corresponding tringle plne, nd the distnce between two prllel plnes is times of the edge length. Besides, ech of the four sets of prllel plnes is perpendiculr to the corresponding digonl of the cube. A truncted octhedron is got by cutting rectngulr pyrmid whose length is of regulr octhedron t the six vertexes of the regulr octhedron. For reserch purposes, the wy of chieving truncted octhedron is chnged into cutting the cube, which is the sme s cutting cubocthedron. Just chnge the cutting in of the edge length into of the edge length, nd we cn chieve truncted octhedron. Its three Orthogrphic Views re the sme, which is the result of cutting in mny points of the cubocthedron nd lot of geometric fetures re similr to the cubocthedron. The squre plne is in prllel to its corresponding squre plne, nd the distnce between the two prllel plnes is times of the edge length; the regulr hexgon is in prllel to its corresponding regulr hexgon, but the distnce between the two prllel plnes is times of the edge length, just s ech of the four sets of prllel plnes is perpendiculr to the corresponding digonl of the cube (See Fig. ). ()Truncted (b) Cubocthedron (c)truncted Tetrhedron Octhedron Figure. Bsic cell. Suppose the edge length is, nd we cn get the following results: the volume of the truncted tetrhedron is, its surfce re is ; the volume of cubocthedron is, nd its surfce re is ( ) ; the volume of truncted octhedron is 8, nd the surfce re is ( ). The volume rtio of truncted tetrhedron, cubocthedron nd truncted octhedron is ::9, while the surfce re rtio is :( + ) :( + ). As shown in Figure, the height of truncted octhedron is, which is twice the height of the cubocthedron. From Tble, we cn see tht the two dihedrl ngles of the truncted tetrhedron re 9. ( cos ),. ( cos ); the dihedrl ngle of the cubocthedron is. ( cos ); the dihedrl. ngles of the truncted octhedron re ( cos ), 9. ( cos ). By trigonometric formul, we cn see the reltionships between these dihedrl ngles:. 9. 8, nd. 9.. As long s the squre surfces of the cubocthedron nd the truncted octhedron cn join closely, the regulr hexgon surfces of the truncted octhedron nd the truncted tetrhedron cn join closely, so cn the regulr tringle of the truncted tetrhedron nd the cubocthedron. Then, the reltionships between the dihedrl ngles stisfy nd 8, nd three kinds of polyhedron combintions cn grow concomitntly nd cn fill spce semlessly. We cn lso see from nother perspective, connect the cubocthedron in the direction of the six squre fces by the truncted octhedron. In this frmework, only the truncted tetrhedron s cvity is left unstudied during the reserch of structures, so we only need to focus on the explortion of the cube cutting out of the cubocthedron nd truncted octhedron. B. Wys of Spce Polyhedrons Combintion nd Bsic Units of Polyhedrons Combintion Suppose there re 8 truncted tetrhedrons, cubocthedrons nd truncted octhedrons, then the polyhedrons cn mke up solidly stcked cubic block of type, compct filling spce just like cube. Suppose truncted octhedron is ttched to cubocthedron (or cubocthedron is ttched to truncted octhedron) s unit, then we cn construct n rry of (see Fig. () nd Fig. (b)). Suppose the sixsix-edge(common edge of two regulr hexgons) of two truncted tetrhedron overlp ech other s one unit of coopertion, then we cn construct n rry of (Fig. (c)). Suppose the former six-six-edge unit rottes 9 degrees round the Z xis, we cn construct n rry of (Fig. (d). Refer to Figure (e) nd combine (), (b), (c), nd (d) twice in Figure, nd then solid cubic block contining polyhedrons is formed(fig. (f)). When the bsic unit of rrys of is set to be combined, we cn get bsic unit combintion of 8 truncted tetrhedrons, cubocthedrons nd truncted octhedrons (Fig. (g)). The bsic unit combintion, just DOI./IJSSST.... ISSN: -8x online, -8 print

4 inscribed in cube, forms compct ccumultion body long three mutully orthogonl directions of the X, Y, nd Z xis, filling three dimensionl spce firmly. () rry of truncted Octhedron nd cubocthedrons (b) rry of cubocthedrons nd truncted Octhedron (c) rry of (d) rry of two two connected truncted tetrhedrons truncted tetrhedrons fter rottion of 9 (e)verticl view of cubic block s bsl fist lyer front view of bsic unit combintion verticl view (f)compct ccumultion side view (g)bsic unit combintion Figure. Formtion of polyhedrons composition nd bsic cell composition. IV. CUTTING A POLYHEDRAL DENSE PACKING BLOCK In the construction prctice of the Wter Cube, Yu Weijing et l. studied the cutting problems of Weire- Pheln polyhedron's pcking in gret detil []. The position of the cutting surfce must pss through the vertex of the polyhedron; otherwise, the types of the surfce chord rod nd bdominl rod would increse substntilly. Wht s more, the minimum distnce from the internl node to the cutting surfce will be shortened nd this is not conducive to meeting the structurl requirements of the structure. In ddition, The djcent cutting surfces with the gretest distnce is the most significnt in the prcticl engineering for the minimum distnce of the cutting surfce from the internl node reches the mximum vlue. A. The Anlysis of Cutting Modes for Polyhedrl Compct Accumultion Bodies In the polyhedrl compct ccumultion cube (Fig.(f)), there is high symmetry in three kinds of polyhedrons, nd the three orthogrphic views of the cubocthedron re consistent with those of the truncted octhedron. The top view of the truncted tetrhedron is consistent with its left view nd is lso congruent with its front view, nd their corresponding plnes re in prllel to ech other. When the compct ccumultion cube doesn t rotte nd the cutting surfce is perpendiculr to the X, Y or Z xis, the cutting surfce ptterns re consistent with ech other in the sme cutting position. Therefore, one only needs to consider the cutting in the direction of just one xis. In the cubocthedron nd the truncted octhedron, four sets of prllel plnes re ll verticl to their corresponding body digonls. When the cutting plne is verticl to the body digonl of the compct ccumultion cube, the cutting surfce ptterns re the sme in the sme cutting position. Therefore, one only needs to whether the cutting surfce is verticl to ny one direction of the four body digonls. When the compct ccumultion cube rottes round the X, Y or Z xis respectively nd the rottion ngle is the sme, the cutting surfce pttern is lso consistent. Therefore, when cutting the surfce, one only needs to consider the condition of rotting round just one coordinte xis. When the cube rottes degrees round the (,, ) to (,, ) vector xis, it will just overlp the originl figure. However, if we cut in the sme position perpendiculr to X, Y or Z xis respectively fter the polyhedrl compct ccumultion cube rottes degrees round the (,,) to (,,) vector xis, the cutting surfce ptterns re consistent. In Figure, the eight vertex coordintes of the originl cube re O(,,), P(,,), A(,,), B(,,), C(,,), D(,,), E(,,), nd F(,,), nd when they rotte degrees round OP vector xis counterclockwise, we cn, DOI./IJSSST.... ISSN: -8x online, -8 print

5 ccording to the spce coordinte rottion formul, get six new vertex coordintes except point O nd point P: A (,, ),B (,, ),C (,, ),D (,, ),E (,, ),nd F (,, ). We cn esily come to the conclusion tht points M, N, S, T, U, nd V re ll the intersection point s well s midpoint of the two cubic edges. Therefore, mking the cross-section prllel to plne XY, plne XZ nd plne YZ fter rotting cube OABC-EDPF degrees round OP vector xis counterclockwise equls plne in prllel with plne PNV, plne PTU nd plne PMS mde in the originl cube. polygons shown in the shded prt of the figure( Fig. (b)): one is squre whose side length is, nother is squre whose side length is nd the lst one is rectngle whose side lengths re nd respectively. Figure. Cutting plnes of polyhedron's pcking without rottion. D z E M F N P C F E V S A x D O T B U C y B A Figure. Cube rotting round the (,,) to (,,) xis with n ngle of. B. The cutting surfce nd its pttern dimensions of the polyhedrl ccumultion solid In order to describe the position nd the size of the cutting surfce precisely, the following is study of the three regulr polyhedrons whose edges re ll supposed to be in length. ) The cutting perpendiculr to the coordinte xes without rottion If the pcking pcked cubic block does not rotte, let s tke the cutting surfce perpendiculr to Z xis (in prllel with the XY plne) s n exmple. In height rnge of bsic unit, there re only 9 cutting surfces through the vertices of the polyhedrons (See Fig. ). We cn clculte tht ll cutting surfces re eqully spced: they re ll. Besides, the minimum distnce between ll the cutting surfces nd the internl nodes re lso equl: they re lso. The surfce ptterns, produced by the cutting surfce positions, nd 9, contin two different polygons shown in the shded prt of the grph (Fig. ()): One is squre whose side length is, nd the other is symmetric octgon whose side lengths re nd respectively. The rest of the surfce ptterns produced by the other cutting surfce positions contin three different () Cutting surfce of (b) cutting surfce of,,9 the other position Figure. Cutting surfce ptterns of polyhedron's pcking without rottion. ) The cutting verticl to the digonl from (,,) to (,,) When the cutting plne is perpendiculr to the digonl of the closely pcked cubic block, let s tke the cutting verticl to the digonl from (,, ) to (,, ) s n exmple. There re only 9 cutting plnes through the vertices of the polyhedrons in the rnge of bsic unit combintion (Fig. ). All cutting surfces re eqully spced: they re ll. So re the minimum distnce between the cutting surfce nd the internl nodes: they re ll. The surfce pttern, produced by the, nd cutting plne, contins only regulr hexgon whose length is (Fig. ()). While the surfce ptterns produced by the, nd 8 cutting plne cn be seen in Fig. (b), the ptterns contin different kinds of polygons (see the shded re of the figure): one is regulr tringle whose length is, one is regulr tringle whose length is, nd the other is hexgon whose lengths re nd respectively. Rotte the produced ptterns of the,, 8 cutting plne 8 degree, nd we cn get the sme ptterns of the,, 9 cutting plne, s they re essentilly the sme. DOI./IJSSST.... ISSN: -8x online, -8 print

6 Figure. Cutting plnes of polyhedron's pcking perpendiculr to the digonl of (,,) (,,). 8 9 which one side is nd nother side is ; nother is similr rectngle of which one side is nd nother side is ; the third one is n isosceles tringle of which the wist is nd the bse is ; the lst one is symmetricl hexgon of which one side is nd nother side is. 8 9 Figure 8. Cutting plnes of polyhedron's pcking rotting through round the X- xis. () Cutting surfce of (b) Cutting surfce of,, the other position Figure. Cutting surfce ptterns of polyhedron's pcking perpendiculr to the digonl of (,,) (,,). ) The cutting round coordinte xes of rottion Suppose the densely pcked cubic block rottes degrees round the X xis, nd Figure 8 is the cutting surfce which strts from the vertex of the polyhedron in the Z direction nd runs prllel with the XY plne fter its rottion of the YZ plne view. In height rnge of bsic unit, there re only cutting surfces through the vertices of the polyhedrons. All cutting surfces re eqully spced: they re ll. Besides, ll the minimum distnces between the cutting surfces nd the internl nodes re the sme, nd they re lso.the surfce ptterns, produced by the cutting surfce positions,,,nd, contin four different polygons shown in the shded prt of the grph (Fig. 9()): Two re isosceles pentgons, which re symmetricl to ech other. Their two long wists re, nd their two short wists re while the bse is ; nother is symmetricl hexgon, of which one side is nd nother side is ; the other is lso symmetricl hexgon, of which one side is nd nother side is. The surfce ptterns, produced by the other cutting surfce positions, contin four different polygons shown in the shded prt of the grph (Fig. 9(b)): One is rectngle of ()cutting surfce of (b) cutting surfce of,,,, the other position Figure 9. Cutting surfce ptterns of polyhedron's pcking rotting through round the X- xis. Suppose the pcking stcked cubic block rottes degrees ( tn ) round the X xis, nd Figure is the cutting surfce which goes through the polyhedron vertices in the Z xis direction nd runs prllel with the XY plne. Besides, Fig. is lso the YZ plne view fter rotting degrees round the X xis. All cutting surfces re eqully spced: they re ll, while ll the minimum distnces between the cutting surfce nd the internl nodes re the sme: they re lso. We cn get five shpes of section polygons by this kind of cutting of the truncted octhedron(see Fig.()): No. is n isosceles tringle of which bse is nd the wist is ; No. is n isosceles pentgon of which the two long wists re, the two short wists re nd the bse is ; No. is n isosceles pentgon of which the two long wists re DOI./IJSSST.... ISSN: -8x online, -8 print

7 , the two short wists re nd the bse is ; No. is n isosceles septngle of which the two wists re, nd respectively while its bse is ; No. is symmetricl hexgon of which the two opposite long sides re ll nd the other four sides re. There re three forms of section polygons for the cubocthedron (see Fig. (b)): No. is n isosceles trpezoid of which the topline nd the bseline re nd respectively, nd the two wists re ; No. is n isosceles pentgon of which the two prllel long wists re re nd the bse is, the two short wists ; No. is symmetricl octgon of which the two opposite long sides re the two opposite short sides re sides re ll., nd the other four The surfce ptterns, produced by the cutting surfce positions, nd 8, contin eight different polygons shown in the shded prt of the grph (see Fig. ()). Of them, there re three section polygons of the truncted octhedron: two section polygons re No. nd the other is No.. Another section polygon is No. of cubocthedron. The other four identicl ones re section polygons of the truncted tetrhedron whose four lengths re nd,, respectively. The surfce ptterns, produced by the other cutting surfce positions, contin 9 different polygons shown in the shded prt of the grph (Fig. (b)). Of them, three section polygons re No., No. nd No. of the truncted octhedron; two section polygons re No. nd No. of cubocthedron; the other four section polygons re two kinds of the truncted tetrhedron which re in pirwise symmetry in pttern: one pttern is tringle whose lengths re, nd respectively; the other is qudrngle whose lengths re,, nd respectively. In Figure (b), the cutting surfce pttern is similr to the ptterns of broken glss. Figure. Cutting plnes of polyhedron's pcking rotting through ( tn ) round the X- xis. 8 9 () section polygons of the truncted octhedron (b) section polygons of the cubocthedron Figure. Section polygons of the truncted octhedron nd cubocthedron rotting through ( tn ) round the X- xis. () cutting surfce of (b) cutting surfce of,,8 the other position Figure. Cutting surfce ptterns of polyhedron's pcking rotting through ( tn ) round the X- xis. After the dense ccumultion cubic block rottes nd degrees round the X xis, the cutting surfce ptterns re the sme, but the distnces between the djcent cutting surfces re very short. Tking the cubic block with bsic unit combintion s component for exmple, the DOI./IJSSST.... ISSN: -8x online, -8 print

8 rtio of the mximum distnce between the djcent cutting surfces nd the rod length is bout /. With the enlrgement of the cubic block, the distnce will become smller nd smller. Therefore, it does not hve the ctul engineering significnce. ) The cutting tht rottes round the vector xis from (,, ) to (,, ) Suppose the pcking stcked cubic block rottes degrees round the (,,) to (,,) vector xis, mking the cutting surfce prllel with the YZ plne through the vertexes of the polyhedrons in the X xis direction is equivlent to mking the cutting surfce prllel with PMS plne in the orl cube(see Fig. ). We cn conclude tht ll cutting surfces re eqully spced by this cutting method: they re ll, nd the minimum distnce between ll 9 the cutting surfces nd the internl nodes re lso 9. Cutting the truncted octhedron in this wy, there re kinds of cutting positions but there re only six ptterns of section polygons while there re kinds of cutting positions cutting the cubocthedron with only four ptterns of section polygons (see Fig. ). The six ptterns of section polygons of the truncted octhedron re s follows (Fig. ()): No. is n isosceles trpezoid of which the topline nd the bseline re nd, nd the wists re ; No. is n isosceles hexgon of which the topline nd the bseline re nd, the two long wists re nd the two short wists re ; No. is n isosceles hexgon of which the topline nd the bseline re nd, the two long wists re nd the two short wists re ; No. is n isosceles hexgon of which the topline nd the bseline re nd respectively, the two long wists re nd the two short wists re ; No. is n isosceles hexgon of which the topline nd the bseline re nd respectively, the two long wists re nd the two short wists re ; No. is symmetric isosceles hexgon of which the topline nd the bseline re both nd the four wists re ll. The four shpes of the section polygons of the cubocthedron re s follows(see Fig. (b)):no. is n isosceles trpezoid of which the topline nd the bseline re nd respectively, nd the wists re ll ; No. is n isosceles hexgon of which the topline nd the bseline re nd 9 8 () cutting surfces of the truncted octhedron respectively, the two long wists re both (b) cutting surfces of the cubocthedron () section polygons of truncted octhedron nd the two short wists re both ; No. is n isosceles hexgon of which the topline nd the bseline re both, the two long wists re both nd the two short wists re both ; No. is symmetric isosceles hexgon of which the topline nd the bseline re both nd the four wists re ll. Figure. Schemtic of cutting plnes of polyhedron rotting through round the (,,) to (,,) vector xis. Figure. Section polygons of the truncted octhedron nd cubocthedron rotting through round the (,,) to (,,) vector xis. Due to the high degree of repetbility of the polyhedron ccumultion block, we only need to consider the cutting in the height rnge of truncted octhedron. The surfce ptterns, produced by the cutting surfce positions, nd 9 shown in Fig. (), include ten different kinds of polygons (shown in the shded prt of the DOI./IJSSST....8 ISSN: -8x online, -8 print

9 grph): Three section polygons re the No. of the truncted octhedron, the reflection symmetry of No. nd the No. of the truncted octhedron; three others re the No. of the cubocthedron, the reflection symmetry of the No. nd the No. of the cubocthedron. the rest four re two kinds of pirwise symmetricl section polygons of the truncted tetrhedron: one is n isosceles tringle of which the bse is nd the two wists re ; the other is n isosceles hexgon of which the topline nd the bseline re nd respectively, the two long wists re nd the two short wists re. The surfce ptterns, produced by the other cutting surfce positions of the truncted octhedron shown in Figure (b), include eleven different polygons (shown in the shded prt of the grph): Four section polygons re No., No., No. nd No. of the truncted octhedron; two section polygons re No. nd No. of the cubocthedron; five section polygons re section polygons of the truncted tetrhedron, including two isosceles tringles, n isosceles trpezoid, nd two isosceles pentgons. For the two isosceles tringles, one is n isosceles tringle of which the wists re nd the bse is, wheres the other is one of which the wists re nd the bse is ; s for the isosceles trpezoid, its topline nd bseline re nd respectively, nd its wists re ; with one of the two isosceles pentgons, its two long wists re, its two short wists re nd its bse is ; with the other one, its two long wists re, its two short wists re nd its bse is. After the densely pcked polyhedrl cube hs rotted degrees round the (,,) to (,,) vector xis, the distnce between the djcent cutting surfces becomes very smll nd when it hs rotted degrees round it becomes even smller. Therefore, the polyhedron spce structures produced by these two kinds of rottion ngles do not hve prcticl engineering significnce. () cutting surfce of,, 9 () cutting surfce of The other position Figure. Cutting surfce ptterns of polyhedron's pcking rotting through round the(,,) to (,,) vector xis. V. A BRIEF SUMMARY The combintion of the regulr truncted tetrhedron, the regulr cubocthedron nd the regulr truncted octhedron cn form regulr closely pcked cubic block. The four wys of cutting, cutting the densely pcked block perpendiculr to the coordinte xes, cutting it perpendiculr to the solid digonl, cutting it round the coordinte xes of rottion, nd cutting it round the vector xis of rottion from (,, ) to (,, ), whose structure dimensions (the cutting surfces spcing) cn meet the needs of the ctul project, nd the ptterns of the cutting surfce show high degree of repetbility nd vriety. For exmple, the Figure (b) resembles the ptterns of broken glss. This study provides some smples for the building elevtions nd the ptterns of roof. Unfortuntely, with the phenomenon, reserchers hve so fr filed to come up with n ide like Beijing s fmous Wter Cube, which produces surfce pttern of rndom nd disorderly visul effect. Besides the spce orgniztion filling model of the truncted tetrhedron, the cubocthedron nd the truncted octhedron, for others semless spce combintion filling models of Pltonic solids nd Archimeden solids, whose ptterns of the cutting surfce cn be so exquisite, whether there exist ny cutting surfce ptterns of rndom but disorderly visul effect like Beijing s fmous Wter Cube bubbles clls for further explortion. In ddition, besides the reserch on the pcking solid s cutting surfce ptterns, the pnel structures nd the frme structures cn be built into right squre pyrmids nd cylinders in vriety of sizes (by mens of cutting) due to the high symmetry of the closely pcked solids discussed in this pper. This structurl function remins to be further studied. This pper only hs studied the geometric constitution of the spce combintion structures composed of the truncted tetrhedron, the cubocthedron nd the truncted octhedron. It is necessry to mke systemtic study nd nlysis further on the stress performnce of the sttic nd dynmic behvior of pnel structures nd frme structures in order to meet the needs of the rel engineering project. DOI./IJSSST....9 ISSN: -8x online, -8 print

10 REFERENCES [] Ed. Weire D. The Kelvin Problem:Fom Structures of Miniml Surfce Are. London: Tylor& Frncis, 99. [] Weire D nd Pheln R. A counter-exmple to Kelvin's conjecture on miniml surfces. Philosophicl Mgzine Letters, vol.9, no., pp.-, 99. [] Linus Puling. Nture of the Chemicl Bond (rd ed). New York: Cornell University Press,9: 8. [] Zhou gong-du, ed. Polyhedr in chemistry[m]. Beijing University Press, Beijing, Chin, 9 (in Chinese). [] Xing Xin-n, Zho Yng, Dong Shi-lin. Geometric constitution of spce frme structure bsed on spce-filling polyhedron. Journl of Zhejing University (Engineering Science), vol., no., pp. -, 8. [] Chen Xin-chun, Zho Yng, Gu Lei, Fu Xue-yi, Dong Shi-lin. Modeling methods for polyhedron spce frme structures. Journl of Zhejing University (Engineering Science), vol.9, no., pp. 9-9,. [] Yu Wei-jing, Zho Yng, Gu Lei, et l. Optimiztion of geometric constitution of new polyhedron sptil frmes. Journl of Building Structures, vol., no., pp. -,. DOI./IJSSST.... ISSN: -8x online, -8 print