document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=367070>Question 367070</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I'm hoping someone can please help me! I'm having issues trying to figure out this problem. Your help is greatly appreciated! Thank you!!!<BR>\n" );
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document.write( "Base of polyhedral	Vertices	Faces	Edges<BR>\n" );
document.write( "Triangle	4vertices	4faces	6edges<BR>\n" );
document.write( "Square	____vertices	____faces	8edges<BR>\n" );
document.write( "Pentagon	____vertices	6faces	____edges<BR>\n" );
document.write( "Hexagon	7vertices	____faces	____edges<BR>\n" );
document.write( "What type of polyhedral is represented in the table?<BR>\n" );
document.write( "If n represents the number of sides of the polyhedra base, then write an equation for the number of vertices, faces, and edges of the polyhedral. \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #261718 by </B><A HREF=/tutors/aboutme.mpl?userid=Jk22><B>Jk22(389)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Jk22><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Jk22><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=261718\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Supposing the problem were to build a polyhedron with the same regular polygon as faces : <BR>\n" );
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document.write( "Taking Euler's formula for polyhedron, we have : <BR>\n" );
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document.write( "V - E + F = 2 (vertex minus edge + face number equals 2)<BR>\n" );
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document.write( "suppose the faces were the same base polygon with n side (n vertices)<BR>\n" );
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document.write( "then : <BR>\n" );
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document.write( " n*F/2 = E (each side of the F polygons (with n sides) are shared by 2 faces)<BR>\n" );
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document.write( " n*F/3 = V (each vertex of the n*F is shared by 3 faces)<BR>\n" );
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document.write( "the equation becomes : n*F/3 - n*F/2 + F = 2<BR>\n" );
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document.write( "or (6-n)F = 12<BR>\n" );
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document.write( "F = 12/(6-n), E = 6n/(6-n), V = 4n/(6-n)<BR>\n" );
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document.write( "triangle : n = 3 : V = 12/3 = 4, E = 18/3 = 6, F = 12/3 = 4<BR>\n" );
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document.write( "square : n = 4 : V = 16/2 = 8, E = 24/2 = 12, F = 12/2 = 6<BR>\n" );
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document.write( "pentagon : n = 5 : V = 20, E = 30, F = 12<BR>\n" );
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document.write( "hexagon : n = 6, all are infinite, this covers the plane.\r<BR>\n" );
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document.write( "if the vertex is shared by m faces : V = n*F/m<BR>\n" );
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document.write( "n*F/m - n*F/2 + F = 2<BR>\n" );
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document.write( "(2m + 2n - nm)F = 4m<BR>\n" );
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document.write( "F = 4m/(m(2-n)+2n)<BR>\n" );
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document.write( "if n=3 (base triangle), F = 4m/(6-m), <BR>\n" );
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document.write( "m-----3----4-----5<BR>\n" );
document.write( "4m----12---16----20<BR>\n" );
document.write( "6-m---3----2-----1<BR>\n" );
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document.write( "F-----4----8-----20\r<BR>\n" );
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document.write( "tetrahedron, octahedron and icosahedron, <BR>\n" );
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document.write( "if n=4 (base square), F = 4m/(8-2m)<BR>\n" );
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document.write( "m=3, F = 12/2 = 6 (cube) (only 3 faces per vertex possible)<BR>\n" );
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document.write( "n=5 (base pentagon), F = 4m/(10-3m), => m=3 dodecahedron<BR>\n" );
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document.write( "other solutions : the number of faces per vertex are mixed (like complementary fullerenes) : suppose base as a triangle<BR>\n" );
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document.write( "suppose n1 and n2 are the possible number of faces per vertex, with 3F/k vertices n2-connected, and 3F(1-1/k) n1-connected<BR>\n" );
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document.write( "F(n2(6-n1) + 6/k*(n1-n2)) = 4n1n2<BR>\n" );
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document.write( "for n1=5, n2=3, : F=12, k=6 : 10 vertices 5-connected, 2 vertices 3-connected<BR>\n" );
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document.write( "for n1=3, n2=5, F=20, k=2, 10 vertices 3-connected, and 10 5-connected<BR>\n" );
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document.write( "if the polyhedron is made out of 2 faces' type, there are also other cases : <BR>\n" );
document.write( "polyhedra called fullerenes\r<BR>\n" );
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document.write( "(hexagons and other polygons, or other mix of polygons).\r<BR>\n" );
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document.write( "the formula becomes : n1*(6-s1) + n2*(6-s2) = 12, where ni are the numbers of regular polygons with si sides.<BR>\n" );
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document.write( "This gives solution like : \r<BR>\n" );
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document.write( " heptagons (2,3,4,6) and pentagons (14, 15, 16, 18)<BR>\n" );
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document.write( " 4 heptagons and 8 squares<BR>\n" );
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document.write( " or heptagons (6) and triangles (6) should be able to build a polyhedron ? </SPAN>\n" );
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