undefined fundamental surface in hyperbolic geometry
in [Math]
|
From: gudi on 28 May 2010 10:55 Two of the 5 Platonic solids can be formed thus: A unit circle has angle 360 deg at center divided into 4/6 parts of 90/60 degrees each. Gauss Curvature positive. One fraction is cut and _removed_ so that 3/5 sectors are respy left. Other corners are made up into vertices of regular polygons ( flat squares and triangles) and when continuation surface is assembled repetitively, form a cube/ icosahedron, inscribable in a sphere as platonic solids with rotational symmerties. Gauss Curvature is negative. Now one extra fraction is added so that 5/7 sectors respy come together at a vertex/center in warped assembly. These are made up into regular polygons ( flat squares and triangles) and when continuation surface is assembled repetitively, form a new hyperbolic surface, embeddable on a beautifully warped surface that has not been described anywhere, imho, upto this point of time. Its existence has also never been investigated. Am I right? I have left out other platonic bodies in the above, but the similar comments apply. With high regards Narasimham
From: Tim Golden BandTech.com on 28 May 2010 11:19 On May 28, 10:55 am, gudi <mathm...(a)hotmail.com> wrote: > Two of the 5 Platonic solids can be formed thus: > > A unit circle has angle 360 deg at center divided into 4/6 parts of > 90/60 degrees each. > > Gauss Curvature positive. > > One fraction is cut and _removed_ so that 3/5 sectors are respy left. > Other corners are made up into vertices of regular polygons ( flat > squares and triangles) and when continuation surface is assembled > repetitively, form a cube/ icosahedron, inscribable in a sphere as > platonic solids with rotational symmerties. > > Gauss Curvature is negative. > > Now one extra fraction is added so that 5/7 sectors respy come > together at a vertex/center in warped assembly. These are made up into > regular polygons ( flat squares and triangles) and when continuation > surface is assembled repetitively, form a new hyperbolic surface, > embeddable on a beautifully warped surface that has not been described > anywhere, imho, upto this point of time. Its existence has also never > been investigated. > > Am I right? Well, partially you are right, but without the folds I have done this: http://bandtechnology.com/ConicalStudy/conic.html I beleive that the Gaussian curvature is zero, except at the singularity, where it is infinitely negative when inserting area. I wish that I could name these 'superplane' and 'subplane', for that is the most pure naming possible. We can build a cone as, say, a 0.523 plane. Likewise we can build a 1.523 plane, or even a 4 plane. Didn't we discuss this a long time ago and you put up some graphics in Mathematica? Maybe that was somebody else. Anyway, I have not attempted to formally publish this work. I congratulate you if you found this on your own. - Tim > > I have left out other platonic bodies in the above, but the similar > comments apply. > > With high regards > > Narasimham
From: gudi on 10 Jun 2010 15:15 On May 28, 8:19 pm, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On May 28, 10:55 am, gudi <mathm...(a)hotmail.com> wrote: > > > Two of the 5 Platonic solids can be formed thus: > > > A unit circle has angle 360 deg at center divided into 4/6 parts of > > 90/60 degrees each. > > > Gauss Curvature positive. > > > One fraction is cut and _removed_ so that 3/5 sectors are respy left.. > > Other corners are made up into vertices of regular polygons ( flat > > squares and triangles) and when continuation surface is assembled > > repetitively, form a cube/ icosahedron, inscribable in a sphere as > > platonic solids with rotational symmerties. > > > Gauss Curvature is negative. > > > Now one extra fraction is added so that 5/7 sectors respy come > > together at a vertex/center in warped assembly. These are made up into > > regular polygons ( flat squares and triangles) and when continuation > > surface is assembled repetitively, form a new hyperbolic surface, > > embeddable on a beautifully warped surface that has not been described > > anywhere, imho, upto this point of time. Its existence has also never > > been investigated. > > > Am I right? > > Well, partially you are right, but without the folds I have done this: > http://bandtechnology.com/ConicalStudy/conic.html > I beleive that the Gaussian curvature is zero, except at the > singularity, where it is infinitely negative when inserting area. I > wish that I could name these 'superplane' and 'subplane', for that is > the most pure naming possible. We can build a cone as, say, a 0.523 > plane. Likewise we can build a 1.523 plane, or even a 4 plane. > > Didn't we discuss this a long time ago and you put up some graphics in > Mathematica? Maybe that was somebody else. Anyway, I have not > attempted to formally publish this work. I congratulate you if you > found this on your own. It is an importantant unexplored field of geometry and there is a big scope for development, imho. > - Tim > > > I have left out other platonic bodies in the above, but the similar > > comments apply. > > > With high regards > > > Narasimham- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - Hi Tim, Sorry for delayed response, it was in part to conceptually gather the related aspects. Indeed we discussed your original post mentioned at that time.. I was also not comfortable with what is generally known or available on the topic, about geometrically including more than 2 Pi angle at vertex ,bending thin paper/sheets under action of varying mechanical bending moment along the edges.I suggested a formulation modifying the Elastica to a conical rather than a cylindrical surface. I attached Mathematica images that resembled what one sees physically by paper bendings on small and extended sectors.In the folowing discussion I take liberty with nomenclature, am sure it is obvious why. How the foliation is taking place can be better seen /described on the surface of a unit sphere rather than focus on what happens at the sphere center/vertex. Accordingly I have shifted the focus of attention to the lines on the sphereical surface. There are four types we can divide them into. 1) Developable/Conoidal foliations. Gauss curvature K is zero. the paper models give central assembly angle exceeeding 2 pi, when the excess over 2 pi is increased, we have the biplane ,triplane you suggested, and more. The vertex subtended angle omega = integral( ds sin(si)/r ) , where s is arc length in spherical coordinates, si is angle between arc s and radius r There are two subdivisions here in smooth paper bendings around vertex which is center of a sphere. Essentially these two types belong to hyperbolic geometry of surfaces , 1a) Wavy and 1b) Intersecting types, are shown in: http://i49.tinypic.com/1zfl7df.jpg 1a) is formed by joining a siniusoidal type of curve drawn on equator of a sphere and connected to sphere center. for 1b), what I call here "intersecting hyperbolic viviani foliations " form, which are just overpacked ordinary cone lateral curved surfaces. For these we note characteristically some essential features: Like the standard cone, the vertex is a singular point. Necessary condition for these surface is K ( Gauss curvature) = 0, so is developable. Difference now is that total length traced by vector extremeties on unit sphere omega must be > 2 pi. ( When equal to 2 pi, a great circle is traced, a standard cone for < 2 pi.) These are 'hyperbolic cones' because the angle subtended at the sphere center/vertex is greater than 2 pi. The Viviani Foliation for nviv =1 should perhaps be recognized as a typical case for a class of hyperbolic cones.For the Viviani curve on a sphere, th = ph, latitude equals longitude, formed by cutting cylinder radius = sphere diameter, cylinder generator passing through a diameter through sphere's poles. The standard ' elliptic cones', if I may use such an apellation, have it less than 2 pi forming on a sphere by joining a latitude circle. In fact the ratio is sin al) = a/R where a is base radius of a latitude circle or cone, R is sphere radius, al is the co- latitude( complement of latitude angle) 1a) and 1b) are isometrically mappable onto a plane, if vertex singularity is overlooked. 2a) Negative K, a Wavy type represented by the generalized monkey saddle: http://i47.tinypic.com/10gjsau.jpg They are represented in Monge form ( z = f (x,y) ) by real ( or imaginary ) part as the harmonic function from (x + i y) ^n where n is a positive integer. E.g., given in http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/genmonkey.html The sign of (r t - s^2) which is the numerator or sign determining part of expression of Monge Form Gauss curvature (discriminant) is indeterminate at the vertex for n >2. 2b) Intersecting non-developable hyperbolic foliation. This is a logical possibility based on truth table completion (wavy/ intersecting/ K > 0, K < 0) of possibilities rather than a speculation. No known formulations, parameterizations or images. ( Entirely imaginary :) at present ). The K = 0 intersecting foliations should be given a curvature to its 'meridians'r u-lines to create these surfaces. In the following Mathematica program, all cases are covered. The classical Viviani curve adaptaion is for nviv =1 ; many hyperbolic foliations occur for nviv =/= 1 values. ph[t_,n_]=.25 Sin[n t]; DevFolio2=ParametricPlot3D[au{Cos[ph[t,2]] Sin[t], Cos[ph[t,2]] Cos[t], Sin[ph[t,2]]},{t,0,2 Pi},{au,.2,1},Mesh->{23,6},PlotRange- >All] DevFolio3=ParametricPlot3D[au{Cos[ph[t,3]] Sin[t], Cos[ph[t,3]] Cos[t], Sin[ph[t,3]]},{t,0,2 Pi},{au,.2,1},Mesh->{23,6},PlotRange- >All] DevFolio8=ParametricPlot3D[au{Cos[ph[t,8]] Sin[t], Cos[ph[t,8]] Cos[t], Sin[ph[t,8]]},{t,0,2 Pi},{au,.2,1},Mesh->{23,6},PlotRange- >All] {x,y}=u {Cos[t],Sin[t]}; Horsesaddle2=ParametricPlot3D[{x,y,u^2 (x^2-y^2)/2},{t,0,2 Pi},{u,. 25,1},PlotRange->All,Mesh->{23,7}] MonkeySad3=ParametricPlot3D[{x,y,u^8 (x^3-3 x y^2)/3},{t,0,2 Pi},{u,. 25,1},PlotRange->All,Mesh->{23,7}] Expand[( x + I y)^8] GenMonkey8=ParametricPlot3D[{x,y,u^8 (x^8-28 x^6 y^2+70 x^4 y^4-28 x^2 y^6+y^8)/5},{t,0,2 Pi},{u,.25,1},PlotRange->All,Mesh->{23,7}] " Viviani type Foliations ; standard Viviani for nviv=1 ; many hyperbolic foliations occur for nviv =/= 1 values " nviv=1; vivxyz={u Cos[nviv*t] Cos[t],u Cos[nviv*t]Sin[t],u Sin[nviv*t]} ; VivianiHypFull=ParametricPlot3D[vivxyz,{u,.1,1},{t,-Pi,Pi},Mesh->{4,12 nviv},PlotRange->All] (*ph,t=latitude,longitude; nviv=1->Bent sheet of Viviani curves*) sphere=ParametricPlot3D[{Cos[ph] Sin[t],Cos[ph] Cos[t],Sin[ph]},{t,Pi +1.3,Pi+1.7},{ph,Pi/2,-Pi/2},Mesh->{4,7}]; Show[{sphere,VivianiHypFull},PlotRange->All,Boxed->False,Axes- >None,PlotLabel->" FOLIATIONS OF VIVIANI WITH nviv = 1 "] nviv=3; vivxyz={u Sin[nviv*t] Cos[t],u Sin[nviv*t]Sin[t],u Cos[nviv*t]} ; TimHyp=ParametricPlot3D[vivxyz,{u,.1,1},{t,-Pi/nviv,Pi/nviv},Mesh- >{4,12},PlotRange->All,PlotLabel->"Tim's sci.math Paper Model, nviv =3 "] TimHypFull=ParametricPlot3D[vivxyz,{u,.1,1},{t,-Pi,Pi},Mesh->{4,12 nviv},PlotRange->All]; (*ph=t=latitude=longitude;nviv=1->Viviani curve*) sphere=ParametricPlot3D[{Cos[ph] Sin[t],Cos[ph] Cos[t],Sin[ph]},{t,Pi +1.35,Pi+1.65},{ph,Pi/2,-Pi/2},Mesh->{4,7}]; Show[{sphere,TimHypFull},PlotRange->All,Boxed->False,Axes- >None,PlotLabel->" Full Foliation of Tim's Paper Model "] (* Mathematica images by G.L.Narasimham *) Tim={ Sin[nviv*t] Cos[t], Sin[nviv*t]Sin[t], Cos[nviv*t]} ; ParametricPlot3D[Tim,{t,-Pi,Pi},PlotLabel->" Tim's Paper Model Boundary on unit sphere "] Next related topic I deal with here is overpacked equilateral triangles at a vertex. Motivation is to cross the upper bound 2 pi on the basis of which the five Platonic solids are created ( at any vertex, angle sum < 2 pi) and thereby seek a smooth surface passing through all the folds to attempt to obtain the case 2b). A situation I have here proposed now has abrupt folds, we consider here flat discrete sectors/triangles instead of smooth bent sheets. the total included angle which is swept out in the developing plane around the vertex is more than 360 deg at each vertex. This " hyperbolic excess angle" can be also seen on laying it flat by radial cuts of the folds on development. The smooth enveloping conoidal surface passing through the folds belong to Viviani surfaces, generalized monkey saddles etc. should be most interesting in hyperbolic geometry ... or so is (my) belief. Whether it is linkable to the surface of constant negative k crocheted by Daina Tamania ( with no known parameterization) is an open question. http://www.math.cornell.edu/~dwh/papers/crochet/crochet.html Many 'hyper Platonic " solids can be made, the total angle at vertex should exceed 360 degrees. Definitely there should be rotational symmetries of these solids. Some possibilities are: 4 sectors of 108 deg of a pentagon coming together at one vertex. ( Three makes a dodecahedron spherical triangle.) 5 sectors of 90 deg squares coming together at one vertex. ( Four make a cube) 7 or 8 equilateral triangles coming together at one vertex.( Four for octahedron, five for icosahedron) Bracketed items form Platonic solids belonging to elliptic geometry. The image below is an " OctaStar" made on ANSYS, Finite Element Analysis structural software, eight equilateral triangles at a vertex. http://i47.tinypic.com/2afzqky.jpg It can be easily constructed from cut cardboard triangles joined together. There are 8 faces, dihedral angle is 99.879366 deg and angle between outer edges is 83.031761 deg. Identical parts of this star can be assembled and a new surface may develop. I do not know to what warped surface definition this sort of space triagulation brings feeding into category 2b). Perhaps you make one model and upload the image for other cases ? Hope that was an interesting discussion. Further critical comments are welcome. With Best Regards Narasimham
From: gudi on 11 Jun 2010 15:07 On May 28, 8:19 pm, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On May 28, 10:55 am, gudi <mathm...(a)hotmail.com> wrote: > > Two of the 5 Platonic solids can be formed thus: > > A unit circle has angle 360 deg at center divided into 4/6 parts of > > 90/60 degrees each. > > Gauss Curvature positive. > > One fraction is cut and _removed_ so that 3/5 sectors are respy left. > > Other corners are made up into vertices of regular polygons ( flat > > squares and triangles) and when continuation surface is assembled > > repetitively, form a cube/ icosahedron, inscribable in a sphere as > > platonic solids with rotational symmerties. > > Gauss Curvature is negative. > > Now one extra fraction is added so that 5/7 sectors respy come > > together at a vertex/center in warped assembly. These are made up into > > regular polygons ( flat squares and triangles) and when continuation > > surface is assembled repetitively, form a new hyperbolic surface, > > embeddable on a beautifully warped surface that has not been described > > anywhere, imho, upto this point of time. Its existence has also never > > been investigated. > > Am I right? > Well, partially you are right, but without the folds I have done this: > http://bandtechnology.com/ConicalStudy/conic.html > I beleive that the Gaussian curvature is zero, except at the > singularity, where it is infinitely negative when inserting area. I > wish that I could name these 'superplane' and 'subplane', for that is > the most pure naming possible. We can build a cone as, say, a 0.523 > plane. Likewise we can build a 1.523 plane, or even a 4 plane. > Didn't we discuss this a long time ago and you put up some graphics in > Mathematica? Maybe that was somebody else. Anyway, I have not > attempted to formally publish this work. I congratulate you if you > found this on your own. > - Tim Carrying forward Platonic solids into the hyperbolic regime is an important but relatively unexplored field of geometry,there is much scope for development, imho. > > I have left out other platonic bodies in the above, but the similar > > comments apply. > > With high regards > > Narasimham- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - Hi Tim, Sorry for delayed response, I was collecting my thoughts related aspects. Indeed we discussed your original post mentioned at that time.. I was also not comfortable with what is generally known or available on the topic, about geometrically including more than 2 Pi angle at vertex ,bending thin paper/sheets under action of varying mechanical bending moment along the edges.(I suggested a formulation modifying the Elastica to a conical rather than a cylindrical surface. I attached Mathematica images that resembled what one sees physically by paper bendings on small and extended sectors). The paper model you posted resembles ( nviv =3, 3 foliations) : http://i47.tinypic.com/10gjsau.jpg The three foliations are not obvious in the first image but is clear when extended domain/range of surface is depicted. In the following discussion I take some liberty with nomenclatures,may be obvious why. How the foliation takes place can be better seen /described on the surface of a unit sphere rather than when we focus on what happens at the sphere center/vertex point of concurrence. Accordingly, I have shifted focus of attention to the lines on the spherical surface, away from vertex. There are four types we can divide the foliation formations into. 1) Developable/Conoidal foliations. Gauss curvature K is zero. the paper models give central assembly angle exceeeding 2 pi, when the excess over 2 pi is increased, we have the biplane ,triplane you suggested, and more. Subtended angle is say omega = integral( ds sin(si)/r ) at vertex , where s is arc length , si is angle between arc s and slant radius r in spherical coordinates. There are two subdivisions here in smooth paper bendings around vertex as center of a sphere. Essentially these two types belong to hyperbolic geometry of surfaces :- 1a) Wavy foliation and 1b) Intersecting foliation types, are shown respectively in first part of image http://i49.tinypic.com/1zfl7df.jpg and in http://i47.tinypic.com/10gjsau.jpg 1a) is formed by joining a siniusoidal type of curve drawn on equator of a sphere and connected to sphere center. for 1b), what I call here "intersecting hyperbolic viviani foliations", are just overpacked ordinary cone lateral curved surfaces. They are formed by joining self intersecting lines on sphere to the sphere center. For these foliations we note characteristically some essential features: Like the standard cone, the vertex is a singular point. Necessary condition for these surface is K ( Gauss curvature) = 0, so is developable. Difference with 1a) now is that total length traced by vector extremeties on unit sphere must be = omega > 2 pi. ( When it equals 2 pi, a great circle is traced, a standard cone for < 2 pi) . These may be called 'hyperbolic cones' because the angle subtended at the sphere center/vertex, omega is greater than 2 pi. The Viviani Foliation for nviv =1 should perhaps be recognized as a typical or central case for a class of hyperbolic cones of omega > 2 pi. For the Viviani curve on a sphere, th = ph, or latitude equals longitude, formed by intersection a cylinder whose radius = sphere diameter, cylinder generator coinciding with a diameter going through sphere's poles. Foliations of case nviv=2 resemble the Enneper surface, but with K =0. (Enneper surface has K< 0). These are all orientable. Non-integer nviv gives rise to infinite, non-repeating foliations. The standard ' elliptic cones', if I may now use such an apellation, have omega less than 2 pi at vertex of a sphere, formed by joining a points of a latitude circle to center of sphere,( i.e., joining points on a small circle to center). In fact, the ratio is sin(al) = a/R where a is base radius of a latitude circle or cone, R is sphere radius ( cone slant radius =1). Semi vertical angle of cone al is the co- latitude ( complement of latitude angle) for a cone with central axis of latitude = pi/2. 1a) and 1b) are isometrically mappable onto a plane, if vertex singularity is overlooked. 2a) Negative Gauss curvature K, non-developable Wavy types represented by the generalized monkey saddle in the second part of above picture http://i49.tinypic.com/1zfl7df.jpg They are represented in Monge form ( z = f (x,y) ) by real ( or imaginary ) part as the harmonic function from (x + i y) ^n where n is a positive integer. E.g., given in http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/genmonkey.html The sign of (r t - s^2) which is the numerator or sign determining part of expression of Monge Form Gauss curvature (discriminant) is indeterminate at the vertex for n >2 ( hypar case). Lastly, 2b) Intersecting non-developable hyperbolic foliation. This is a logical possibility based on 4 possibilities (wavy/ intersecting/ K > 0, K < 0), a probability rather than a speculation. No known formulations to me, parameterizations or images. ( Entirely imaginary :) at present ). The K = 0 intersecting foliations should be modified in curvature of its 'meridians'r u-lines to create these surfaces. Appearance of positive K is a problem. In the following Mathematica program, all three cases are covered. The classical Viviani curve adaptaion is for nviv =1 ; many hyperbolic foliations occur for nviv =/= 1 values. ph[t_,n_]=.25 Sin[n t]; DevFolio2=ParametricPlot3D[au{Cos[ph[t,2]] Sin[t], Cos[ph[t,2]] Cos[t], Sin[ph[t,2]]},{t,0,2 Pi},{au,.2,1},Mesh->{23,6},PlotRange- >All] DevFolio3=ParametricPlot3D[au{Cos[ph[t,3]] Sin[t], Cos[ph[t,3]] Cos[t], Sin[ph[t,3]]},{t,0,2 Pi},{au,.2,1},Mesh->{23,6},PlotRange- >All] DevFolio8=ParametricPlot3D[au{Cos[ph[t,8]] Sin[t], Cos[ph[t,8]] Cos[t], Sin[ph[t,8]]},{t,0,2 Pi},{au,.2,1},Mesh->{23,6},PlotRange- >All] {x,y}=u {Cos[t],Sin[t]}; Horsesaddle2=ParametricPlot3D[{x,y,u^2 (x^2-y^2)/2},{t,0,2 Pi},{u,. 25,1},PlotRange->All,Mesh->{23,7}] MonkeySad3=ParametricPlot3D[{x,y,u^8 (x^3-3 x y^2)/3},{t,0,2 Pi},{u,. 25,1},PlotRange->All,Mesh->{23,7}] Expand[( x + I y)^8] GenMonkey8=ParametricPlot3D[{x,y,u^8 (x^8-28 x^6 y^2+70 x^4 y^4-28 x^2 y^6+y^8)/5},{t,0,2 Pi},{u,.25,1},PlotRange->All,Mesh->{23,7}] " Viviani type Foliations ; standard Viviani for nviv=1 ; many hyperbolic foliations occur for nviv =/= 1 values " nviv=1; vivxyz={u Cos[nviv*t] Cos[t],u Cos[nviv*t]Sin[t],u Sin[nviv*t]} ; VivianiHypFull=ParametricPlot3D[vivxyz,{u,.1,1},{t,-Pi,Pi},Mesh- >{4,12 nviv},PlotRange->All] (*ph,t=latitude,longitude; nviv=1->Bent sheet of Viviani curves*) sphere=ParametricPlot3D[{Cos[ph] Sin[t],Cos[ph] Cos[t],Sin[ph]},{t,Pi +1.3,Pi+1.7},{ph,Pi/2,-Pi/2},Mesh->{4,7}]; Show[{sphere,VivianiHypFull},PlotRange->All,Boxed->False,Axes- >None,PlotLabel->" FOLIATIONS OF VIVIANI WITH nviv = 1 "] nviv=3; vivxyz={u Sin[nviv*t] Cos[t],u Sin[nviv*t]Sin[t],u Cos[nviv*t]} ; TimHyp=ParametricPlot3D[vivxyz,{u,.1,1},{t,-Pi/nviv,Pi/nviv},Mesh- >{4,12},PlotRange->All,PlotLabel->"Tim's sci.math Paper Model, nviv =3 "] TimHypFull=ParametricPlot3D[vivxyz,{u,.1,1},{t,-Pi,Pi},Mesh->{4,12 nviv},PlotRange->All]; (*ph=t=latitude=longitude;nviv=1->Viviani curve*) sphere=ParametricPlot3D[{Cos[ph] Sin[t],Cos[ph] Cos[t],Sin[ph]},{t,Pi +1.35,Pi+1.65},{ph,Pi/2,-Pi/2},Mesh->{4,7}]; Show[{sphere,TimHypFull},PlotRange->All,Boxed->False,Axes- >None,PlotLabel->" Full Foliation of Tim's Paper Model "] (* Mathematica images by G.L.Narasimham *) Tim={ Sin[nviv*t] Cos[t], Sin[nviv*t]Sin[t], Cos[nviv*t]} ; ParametricPlot3D[Tim,{t,-Pi,Pi},PlotLabel->" Tim's Paper Model Boundary on unit sphere "] Next we come to the main topic of the thread.We deal with overpacked equilateral triangles at a vertex. Motivation is to cross the upper bound 2 pi on the basis of which the five Platonic solids ( at any vertex, angle sum < 2 pi) are created and thereby attempt to seek a smooth surface passing through all the folds and try obtain the case 2b). A simple situation I propose/show here has abrupt folds, we consider here flat discrete sectors/triangles instead of smooth bent sheets. the total included angle which is swept out in the developing plane around the vertex is more than 360 deg at each vertex. This " hyperbolic excess angle" can be also seen on laying it flat by radial cuts of the folds on development. The smooth enveloping conoidal surface passing through the all the folds belong to Viviani foliations, generalized monkey saddles etc. should be most interesting in hyperbolic geometry ... or so is (my) belief. Whether it is linkable to the surface of constant negative K crocheted together (stitched in) by Daina Tamania ( with no known parameterization for a psedosphere) is an open question. The pattern of lace stiching shows how she incorporated more than 360 degrees at the indicated vertices: http://www.math.cornell.edu/~dwh/papers/crochet/crochet.html Many ' hyperbolic Platonic " solids can be made thus, the total angle at vertex should exceed 360 degrees. Whether such solids are finite in number, is unknown. Definitely there should be rotational symmetries of these hyperbolic solids. Some possibilities are: 4 sectors of 108 deg of a pentagon coming together at one vertex. ( Three make a dodecahedron spherical triangle.) 5 sectors of 90 deg squares coming together at one vertex. ( Three make a cube) 7 or 8 equilateral triangles coming together at one vertex.( Four for octahedron, five for icosahedron) Bracketed items form Platonic solids belonging to elliptic geometry. The image below is an " OctaStar" made on ANSYS, Finite Element Analysis structural software, eight equilateral triangles at a vertex, last case above. http://i47.tinypic.com/2afzqky.jpg It can be easily constructed from cut cardboard triangles joined together. There are 8 faces, dihedral angle is 99.879366 deg and angle between outer edges is 83.031761 deg. Copies of the star can be assembled and a new surface may develop. I do not know to what warped surface definition this sort of space triagulation brings in, after smoothing out to feed into category 2b). Perhaps you or some cares to make one cardboard model and upload the image for other cases ? Hope that was an interesting discussion. All further critical comments are welcome. With Best Regards Narasimham
|
Pages: 1 Prev: Diophantine equation question Next: Interesting optimization problem |