The Platonic solids The five regular polyhedra

Transcription

1 The Platoic solids The five regular polyhedra Ole Witt-Hase jauary 7

2 Cotets. Polygos.... Topologically cosideratios.... Euler s polyhedro theorem.... Regular ets o a sphere.... The dihedral agle i a regular pyramid.... Evaluatio of the dihedral agles ad the radii i the iscribed ad circumscribed spheres of the five regular polyhedra...6. The Tetrahedro Surface ad the volume of the tetrahedro...8. The Cube...9. The Octahedro Surface ad the volume of the octahedro.... The Dodecahedro..... Surface ad the volume of the dodecahedro.... The Icosahedro..... Surface ad the volume of the icosahedro...

3 The five regular polyhedra. Polygos A polygo is a closed curve, where the border cosists of straight lie segmets. Well kow examples are triagles, rectagles ad petagos. The straight lie segmets are the sides or edges of the polygo, ad the itersectio of sides are the corers or the agles of the polygo. The polygo is covex if oe of the agles are greater tha 8. The sum of the agles i a -sided polygo is ( - 8 or ( - π. This is straightforward to verify, oce you kow that the sum of the agles i a triagle is 8. That this is actually the case ca be verified from the two figures below. We oly have to apply three axioms (from Euclid s elemets.. Whe two parallel lies are itersected by a third lie, the correspodig agles of itersectio are equal.. For a give lie ad a give poit outside the lie, you may oly draw oe lie through the poit parallel to the give lie.. Whe two lies itersect, the opposite agles at the itersectio poit are equal. The figure shows a triagle ABC, which is redraw, ow with a lie through B, parallel to AB (axiom. By applyig axiom ad, we ca verify that the three agles above B are equal to the three agles i the triagle, ad their sum is 8. Sice a -sided polygo may always be divided ito a - side polygo ad a triagle the formula ( - 8 for the sum of the agles i a -sided polygo follows. A polygo is regular if all sides are equal ad all agles are equal. The agle i a regular polygo with corers is therefore. ( 8 6 (. v 8 ( Which shows e.g. That the agle i a regular triagle is ( 6, ad the agle i a regular petago is ( 8. A polyhedro is a closed solid, where the surface is built by regular polygos. The simplest polyhedro is the tetrahedro, cosistig of four regular triagles.. Topologically cosideratios The famous polyhedro theorem, first discovered by Euler is most ofte prove i topology.

4 The five regular polyhedra So we wish to map the surface of the polyhedro to a sphere, which circumvets the polyhedro. This will result i a closed et o the sphere, cosistig of masks (m, correspodig to the polygos i the polyhedro, cords (s correspodig to edges of the polygos ad kots (k correspodig to the corers of the polyhedra. We are iterested i trasformig the topological properties of the et to the same topological properties of the polyhedro. Topological properties are properties, that are ivariat upo a cotiuous oe-to-oe mappig (fuctio (ijective mappig. Such a fuctio is called a homeomorphism. Ijectivity of a mappig f from a set A ito a set B implies that to each elemet x i A correspods exactly oe elemet y = f(x i B (f is a fuctio, ad that to each elemet y i B correspods exactly oe elemet x = f - (x i A. Two sets which are geerated by each other by a homeomorphism are said to be topologically equivalet. A simple sectio of a curve is topologically equivalet to a straight lie sectio. Ay closed curve is topologically equivalet to a circle, ad ay piece of a flat sectio is topologically equivalet to a circular disc. Below are show three figures of which all of them are topologically equivalet to a circle. O the figure below E, F ad Y are topologically equivalet, while G ad Z are topologically equivalet, ad W ad X are topologically equivalet.. Euler s polyhedro theorem By a et o a sphere we uderstad a system cosistig of a fiite umber of flat pieces ad a fiite umber of curve pieces, which are deoted masks ad cords respectively, which fulfil the followig coditios.. Every poit o the sphere belogs to at least oe mask (iclusive its border, ad two differet masks do ot have ay iteral poits i commo.. The border of each mask is composed of cords. A poit lyig o a cord is ot a iteral poit, (that is, ot a ed poit of ay other cord or of ay mask

5 The five regular polyhedra To the left is show a et o a sphere, cosistig of masks, cords ad kots. A kot is the ed poit of a cord. The coditios, imply that o the border of each mask i a et there exits at least two kots, ad that every kot i a et lies o the border of exactly two masks. We shall first cosider a simple et cosistig of oly two masks, as show o the secod figure. The oe mask is the oe that you see, ad the secod mask is what is left over o the sphere. Both masks (m are cofied by the same cords (s, ad the same kots (k. Furthermore s = k, sice each cord coects two kots, ad each kot has two cords attached to it. Sice s =k, ad there are two masks, the followig relatio must hold for this most simple et. (. m + k s = I the ext figure we have created a ew mask by drawig a cord with p kots betwee A ad B, i the ier of the mask. I this maer we have added p kots, mask ad p+ cords, which meas that (. is uchaged, as demostrated below. m + + k +p (s + p + = m + k s = I the last figure, we have doe the same, but ow where the cord from A to B lies outside the mask, but it is easily coceived that the result (. is uchaged. What we realize is, that wheever we add a mask to a et, subdued to the coditios, the equatio (. is uchaged. We may the create the etire et (o the figure to the left by addig the masks show, oe by oe. The relatio (. will hold for the complete et, ad therefore for ay et o a sphere that comply to the coditios. If a sphere ecloses a polyhedro, the a cetral projectio of the polyhedro from the cetre of the sphere to the surface of the sphere will be a homeomorphism, which coserves the topology. The sides i the polyhedro, the corers of the polyhedro ad the flat pieces i the polyhedro will correspod to the cords, kots ad the masks of the et with respect to their umbers ad topological relatios. Cosequetly the same relatios betwee sides, corers ad flat pieces (polygos of the polyhedro will hold true as for the cords, kot ad masks of the et. This leads to Euler s Polyhedro theorem. For ay polyhedro, that is homeomorph with a et o a sphere, holds true that the umber of flat pieces (polygos (m, plus the umber of corers (k, mius the umber of cords(s are equal to. (. m + k s =

6 The five regular polyhedra. Regular ets o a sphere A et o a sphere is called regular if there exits two umbers λ ad μ, both beig at least, ad such that, from each kot leaves λ cords, ad the border of each mask cosists of μ cords. We cosider a regular et o a sphere, where the masks (m, kots (k ad cords (s bear the same meaig as before. Sice every mask is limited by μ cords, ad each cord belog to two masks, the followig must hold. m k s ad s Which we rewrite as. m s ad s k Whe isertig these expressios ito Euler s polyhedro theorem m + k s =, we get: Ad by divisio by s: (. s s s s Leadig to the iequality: From which follows that both umbers λ ad μ caot be greater tha, as Furthermore it follows, that oe of the umbers λ ad μ ca be greater tha, sice they are at least ad: 6 A polyhedro is regular, if it cosists of cogruet regular polygos, so that all agles ad all sides i the polygo are equal. For a regular polyhedro, oe ca easily covice oeself that there are oly possibilities for a regular polyhedro, show below i the table (. λ μ s m k polyhedro 6 Tetrahedro 6 8 Cube Dodecahedro 8 6 Octahedro Icosahedro

7 The five regular polyhedra. The dihedral agle i a regular pyramid Aimig at calculatig the dihedral agle, that is, the agle betwee two adjacet polygos which form the regular polyhedro, we shall first fid the dihedral agle i a regular pyramid. Cocerig the tetrahedro, cube ad octahedro this ca be doe usig plae geometry, but for the dodecahedro ad the icosahedro, it is ot possible, sice I have foud o referece to the issue i the mathematical literature. I have however foud a referece to the subject i a mathematical textbook used for the. year of the Daish 9 grade high school (gymasium, but it ivolves spherical geometry i a fairly advaced level. The trick is to itersect a corer at the base of a pyramid by a sphere, havig its cetre at a corer of the base of the pyramid. The poit of doig this is of course that ay corer of a polyhedro ca be cut of to form a regular pyramid. I the two drawigs below is show a corer of a regular pyramid, havig -corer surfaces ad - edges. The top agle i the pyramid is deoted v ad the dihedral agle is φ. As the corer surfaces are isosceles, the agle betwee a side i the corer surface ad a side i the base is ( v v. From the corer O is draw a sphere that itersects the pyramid i the two corer surfaces as well as i the base. So the sphere itersects the pyramid i the two great circle arcs AC ad BC, ad it itersects the base i the great circle arc AB, which accordig to (. is equal to AB =. The arcs AC = BC = v are the agles i the isosceles triagle, which has the top agle v. The arcs AC, BC, AB form a spherical triagle with AC = v, BC = v, AB =, ad where C = φ is the dihedral agle of the pyramid. The dihedral agle ca the be foud from the spherical triagle ABC, where we first write the cosie-relatio i the familiar form, usig the sides a, b, c, correspodig to the agles A, B, C. cos c cos a cosb si asi b cosc cos AB cos BC cos AC si BC si AC cos cos( cos( vcos( v si( vsi( v v cos( si ( cos ( cos v cos( si ( cos v cos ( v cos Spherical Geometry

8 The five regular polyhedra 6 cos From the formula: si, derived from cos x si x, we get whe isertig the expressio for cos φ. si v cos( si ( ( v cos ( v cos ( cos( cos v ( si ( v cos( cos ( v Where we have applied the relatio: cos x si x, ad the formula: cos x cos x cos x cos x I the umerator of the square root. Thus we fid: (. cos( si cos( v. Evaluatio of the dihedral agles ad the radii i the iscribed ad circumscribed spheres of the five regular polyhedra The dihedral agle is the agle betwee two adjacet polygos. If oe cuts a corer of a polyhedro alog the edges of the corer oe has a regular pyramid, where the top agle is the agle i the regular polygos, that costitute the polyhedro, ad where the sides i the base i the pyramid ca be foud from the regular polygo, ad where the pyramid has the same dihedral agle as the regular polyhedro! This is illustrated below for a dodecahedro ad a icosahedro. We start out from (. cos( si cos( v the umber of sides i the pyramid, beig same as the umber of edges from a corer of a regular polyhedro, ad as we i sectio deoted by λ. The umber of corers i the polygos that costitutes the polyhedro we have previously deoted μ.

9 The five regular polyhedra 7 I the formula (. ½v is half the top agle i the pyramid, beig the same as the agle i the regular polygo that costitutes the polyhedro Accordig to (. v. With these otatios, we fid for the half of the dihedral agle: cos( (. si cos( => si cos( si( The values for λ ad μ are give i table (., which we for coveiece repeat below. λ μ s m k polyhedro 6 Tetrahedro 6 8 Cube Dodecahedro 8 6 Octahedro Icosahedro. The Tetrahedro Accordig to the table above (, (,, ad we fid for the dihedral agle. (. si tetrahedro cos( si( 7,,7 The dihedro ad the radii for the iscribed ad circumscribed spheres may relatively easy be calculated usig plae trigoometry. Below is show a tetrahedro, where some lies have bee draw to that purpose. To the right, the triagle ABC has bee extracted, which is used to calculate the radii of the iscribed ad circumscribed spheres. For symmetry reasos the heights i a regular triagle are also the medias of the triagle. If you draw the height from a corer, the the bottom poit will lie i the itersectio poit of the heights i the base triagle.

10 The five regular polyhedra 8 The cetres for the iscribed ad circumscribed spheres, will lie i the commo itersectio of the three heights from the corers. For simplicity we shall put the legth of the side i the polyhedra equal to. The height i the regular triagles ca be foud from Pythagoras: h FB ( From ΔABF we fid the dihedral agle, by drawig the height from B to the opposite edge at G. AG (. si. 7 tetrahedro 7. AB To determie the radii r ad R of the iscribed ad circumscribed spheres, we apply the two oe agled triagles ABC ad AOD. Here AB h ad sice the itersectio poit of the medias divides the media i the fractio : the: AD AB. 6 Furthermore: AC AF FC ( From the two oe agled triagles ABC ad AOD, we thus fid, sice R : r :. (. AO AB AD AC 6 r R 8 The last equatio because O divides AC i the same proportio as D divides AB. R.. Surface ad the volume of the tetrahedro The plae surface ad the volume of the tetrahedro, ca rather easily be determied, but this time we put the legth of the edge to a. R The area of the regular triagle is T ah a, ad the surface of the triagles is therefore: O tetrahedro a The volume of a pyramid is as it is well kow / height of the pyramid times the area of the base. We have the height AC a, ad the volume becomes: V tetrahedro a a a

11 The five regular polyhedra 9. The Cube The dihedral agle of the cube is 9, which also ca be calculated from the formula (. (. si cos( cos( si with (, (,. cos( 9 cube si( We put the legth of the side i the polygos to. I this case it is very easy to determie the radii r ad R of the iscribed ad circumscribed spheres, sice their cetre is the cetre of symmetry, so the radius of the iscribed sphere is. (.6 r The radius of the circumscribed sphere is simply half the diagoal BE. The diagoal of the square is: BH, ad therefore: BE BH EH, so (.7 The surface ad volume of the cube are of course: O Cube = 6a ad V Cube = a R BE. The Octahedro O the figure to the left is show a octahedro. We wish to determie the dihedral agle φ ad the radii r ad R of the iscribed ad circumscribed spheres. I the figure to the left, we have draw the height h from E o BC i the regular triagle BCE, ad we have draw the height h from E o ABCD. The cetre O the iscribed ad circumscribed spheres, lies for symmetry reasos i the cetre of the square ABCD. We fid the half dihedral agle as the agle OFE. EF ( ad Thus we fid: OF OF (.8 cos si 9, 8 9 EF This may be compared with the formula (. cos( si si( med (, (,.

12 The five regular polyhedra si cos( si( Radius i circumscribed sphere is simply half the diagoal of the square ABCD. (.9 R Radius r i the iscribed sphere ca be determied from the triagle OFG, where r = OG, sice the poit of cotact with the sides must be placed i the itersectio poit of the medias (ad the heights of the regular triagle. We therefore have: FG EF. We the get from the right agle triagle OFG: (. r ( ( Surface ad the volume of the octahedro The surface of the octahedro is simply 8 times the area of a regular triagle ad the volume of the octahedro is times the volume of a regular -sided pyramid with base area a. The height of the pyramid equals the radius i the circumscribed sphere. O octahedro. The Dodecahedro 9 8 a a ad V octahedro =,8 a a

13 The five regular polyhedra As remarked earlier, there is probablyo way to fid the dihedral agle, ad cosequetly the radii i the iscribed ad circumscribed spheres from elemetary trigoometry. We therefore resort to formula (.. cos( si with (, (, si( cos( (. si v v 6, 6. si( To determie radii i the iscribed ad circumscribed spheres, we eed to kow the distace r from the cetre of the iscribed ad circumscribed circle of the regular petago to the side, ad the distace R from this cetre to a corer of the petago. As previously, we put the side of the petago to. ( 8 The agle of the petago is 8, ad half the agle is therefore. The cetre agle to a side is therefore 7, ad its half is 6. From the drawig of the petago at the top to the right is see: R (.8 og r ta (.69 si 6 si 6 Radius i the iscribed sphere is determied by the. figure above to the left, which shows a sectio of the dodecahedro, with two petagos itersectig with the dihedral agle. We have draw the two perpediculars from the cetre O for the sphere o the two petagos havig the commo edge DE. The agle ACB is equal to the dihedral agle v of the dodecahedro ad the agle O is O =8 v. The agle AOC is therefore 9 - ½v. The radius r = OA=OB, ad from the triagle AOC is see (. r r ta( 9 v r r ta v (.69 ta 8,. From the triagle ADO, show at the figure to the right, we fid: (. R r r (... Surface ad the volume of the dodecahedro The plae surface of the dodecahedro ca be determied as times the area of a regular petago which is times a triagular sectio with the side a: r a a a ta T petago O dodecahedro ta a I the same maer, the volume determied as times the volume of a regular petago pyramid with a height equal to the radius of the iscribed sphere.

14 The five regular polyhedra V dodecahedro a ar a ta r ta. The Icosahedro As it is the case with the dodecahedro, it is ot possible to determie the dihedral agle of the icosahedro from pure geometry. We therefore use the formula (.. cos( si si( med (, (,. cos( v cos 6 (. si v 8.. si( si 6 Cocerig the radii of the iscribed ad circumscribed spheres, we shall apply the same method as we did for the dodecahedro. r = AC is the radius of the iscribed circle i the regular triagle. r = h 6. R AD is the radius i the circumscribed circle i the regular triagle. From the triagle OAC is see: v r (. r r si(9 r (.8 v cos Ad from the triagle OAD is see: (.6 R OA ( h r. 88 R h.. Surface ad the volume of the icosahedro The plae surface ad the volume of the icosahedro are determied i the same maer as for the dodecahedro, as times the area of the regular triagle, ad times the volume of a sided pyramid with the height equal to the radius of the iscribed sphere. O V a i cos ahedro a i cos ahedro r a a a a v v cos cos

15 The five regular polyhedra V i cos ahedro 9 6 cos This eds our treatmet of the five regular polyhedra. v a