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diff --git a/hacks/glx/polyhedra.c b/hacks/glx/polyhedra.c
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+/*****************************************************************************
+ * #ident "Id: main.c,v 3.27 2002-01-06 16:23:01+02 rl Exp "
+ * kaleido
+ *
+ *	Kaleidoscopic construction of uniform polyhedra
+ *	Copyright (c) 1991-2002 Dr. Zvi Har'El <rl@math.technion.ac.il>
+ *
+ *	Redistribution and use in source and binary forms, with or without
+ *	modification, are permitted provided that the following conditions
+ *	are met:
+ *
+ *	1. Redistributions of source code must retain the above copyright
+ *	   notice, this list of conditions and the following disclaimer.
+ *
+ *	2. Redistributions in binary form must reproduce the above copyright
+ *	   notice, this list of conditions and the following disclaimer in
+ *	   the documentation and/or other materials provided with the
+ *	   distribution.
+ *
+ *	3. The end-user documentation included with the redistribution,
+ *	   if any, must include the following acknowledgment:
+ *		"This product includes software developed by
+ *		 Dr. Zvi Har'El (http://www.math.technion.ac.il/~rl/)."
+ *	   Alternately, this acknowledgment may appear in the software itself,
+ *	   if and wherever such third-party acknowledgments normally appear.
+ *
+ *	This software is provided 'as-is', without any express or implied
+ *	warranty.  In no event will the author be held liable for any
+ *	damages arising from the use of this software.
+ *
+ *	Author:
+ *		Dr. Zvi Har'El,
+ *		Deptartment of Mathematics,
+ *		Technion, Israel Institue of Technology,
+ *		Haifa 32000, Israel.
+ *		E-Mail: rl@math.technion.ac.il
+ *
+ *	ftp://ftp.math.technion.ac.il/kaleido/
+ *	http://www.mathconsult.ch/showroom/unipoly/
+ *
+ *	Adapted for xscreensaver by Jamie Zawinski <jwz@jwz.org> 25-Apr-2004
+ *
+ *****************************************************************************
+ */
+
+#ifdef HAVE_CONFIG_H
+# include "config.h"
+#endif
+
+#include <math.h>
+#include <stdio.h>
+#include <ctype.h>
+#include <string.h>
+#include <stdlib.h>
+#include <errno.h>
+
+#include "polyhedra.h"
+
+extern const char *progname;
+
+#ifndef MAXLONG
+#define MAXLONG 0x7FFFFFFF
+#endif
+#ifndef MAXDIGITS
+#define MAXDIGITS 10 /* (int)log10((double)MAXLONG) + 1 */
+#endif
+
+#ifndef DBL_EPSILON
+#define DBL_EPSILON 2.2204460492503131e-16
+#endif
+#define BIG_EPSILON 3e-2
+#define AZ M_PI/7  /* axis azimuth */
+#define EL M_PI/17 /* axis elevation */
+
+#define Err(x) do {\
+	fprintf (stderr, "%s: %s\n", progname, (x)); \
+	exit (1); \
+    } while(0)
+
+#define Free(lvalue) do {\
+	if (lvalue) {\
+	    free((char*) lvalue);\
+	    lvalue=0;\
+	}\
+    } while(0)
+
+#define Matfree(lvalue,n) do {\
+	if (lvalue) \
+	    matfree((char*) lvalue, n);\
+	lvalue=0;\
+    } while(0)
+
+#define Malloc(lvalue,n,type) do {\
+	if (!(lvalue = (type*) calloc((n), sizeof(type)))) \
+	    abort();\
+    } while(0)
+
+#define Realloc(lvalue,n,type) do {\
+	if (!(lvalue = (type*) realloc(lvalue, (n) * sizeof(type)))) \
+	    abort();\
+    } while(0)
+
+#define Calloc(lvalue,n,type) do {\
+	if (!(lvalue = (type*) calloc(n, sizeof(type))))\
+	    abort();\
+    } while(0)
+
+#define Matalloc(lvalue,n,m,type) do {\
+	if (!(lvalue = (type**) matalloc(n, (m) * sizeof(type))))\
+	    abort();\
+    } while(0)
+
+#define Sprintfrac(lvalue,x) do {\
+	if (!(lvalue=sprintfrac(x)))\
+	    return 0;\
+    } while(0)
+
+#define numerator(x) (frac(x), frax.n)
+#define denominator(x) (frac(x), frax.d)
+#define compl(x) (frac(x), (double) frax.n / (frax.n-frax.d))
+
+typedef struct {
+    double x, y, z;
+} Vector;
+
+typedef struct {
+  /* NOTE: some of the int's can be replaced by short's, char's,
+     or even bit fields, at the expense of readability!!!*/
+  int index; /* index to the standard list, the array uniform[] */
+  int N; /* number of faces types (atmost 5)*/
+  int M; /* vertex valency  (may be big for dihedral polyhedra) */
+  int V; /* vertex count */
+  int E; /* edge count */
+  int F; /* face count */
+  int D; /* density */
+  int chi; /* Euler characteristic */
+  int g; /* order of symmetry group */
+  int K; /* symmetry type: D=2, T=3, O=4, I=5 */
+  int hemi;/* flag hemi polyhedron */
+  int onesided;/* flag onesided polyhedron */
+  int even; /* removed face in pqr| */
+  int *Fi; /* face counts by type (array N)*/
+  int *rot; /* vertex configuration (array M of 0..N-1) */
+  int *snub; /* snub triangle configuration (array M of 0..1) */
+  int *firstrot; /* temporary for vertex generation (array V) */
+  int *anti; /* temporary for direction of ideal vertices (array E) */
+  int *ftype; /* face types (array F) */
+  int **e; /* edges (matrix 2 x E of 0..V-1)*/
+  int **dual_e; /* dual edges (matrix 2 x E of 0..F-1)*/
+  int **incid; /* vertex-face incidence (matrix M x V of 0..F-1)*/
+  int **adj; /* vertex-vertex adjacency (matrix M x V of 0..V-1)*/
+  double p[4]; /* p, q and r; |=0 */
+  double minr; /* smallest nonzero inradius */
+  double gon; /* basis type for dihedral polyhedra */
+  double *n; /* number of side of a face of each type (array N) */
+  double *m; /* number of faces at a vertex of each type (array N) */
+  double *gamma; /* fundamental angles in radians (array N) */
+  char *polyform; /* printable Wythoff symbol */
+  char *config; /* printable vertex configuration */
+  char *group; /* printable group name */
+  char *name; /* name, standard or manifuctured */
+  char *dual_name; /* dual name, standard or manifuctured */
+  char *class;
+  char *dual_class;
+  Vector *v; /* vertex coordinates (array V) */
+  Vector *f; /* face coordinates (array F)*/
+} Polyhedron;
+
+typedef struct {
+  long n,d;
+} Fraction;
+
+static Polyhedron *polyalloc(void);
+static Vector rotate(Vector vertex, Vector axis, double angle);
+
+static Vector sum3(Vector a, Vector b, Vector c);
+static Vector scale(double k, Vector a);
+static Vector sum(Vector a, Vector b);
+static Vector diff(Vector a, Vector b);
+static Vector pole (double r, Vector a, Vector b, Vector c);
+static Vector cross(Vector a, Vector b);
+static double dot(Vector a, Vector b);
+static int same(Vector a, Vector b, double epsilon);
+
+static char *sprintfrac(double x);
+
+static void frac(double x);
+static void matfree(void *mat, int rows);
+static void *matalloc(int rows, int row_size);
+
+static Fraction frax;
+
+
+static const struct {
+  char *Wythoff, *name, *dual, *group, *class, *dual_class;
+  short Coxeter, Wenninger;
+} uniform[] = {
+
+  /****************************************************************************
+   *		Dihedral Schwarz Triangles (D5 only)
+   ***************************************************************************/
+
+  /*  0 */	{"2 5|2",	"Pentagonal Prism",
+				"Pentagonal Dipyramid",
+				"Dihedral (D[1/5])",
+				"",
+				"",
+				0, 0},
+
+  /*  2 */	{"|2 2 5",	"Pentagonal Antiprism",
+				"Pentagonal Deltohedron",
+				"Dihedral (D[1/5])",
+				"",
+				"",
+				0, 0},
+							 /* (2 2 5/2) (D2/5) */
+  /*  4 */	{"2 5/2|2",	"Pentagrammic Prism",
+				"Pentagrammic Dipyramid",
+				"Dihedral (D[2/5])",
+				"",
+				"",
+				0, 0},
+
+  /*  6 */	{"|2 2 5/2",	"Pentagrammic Antiprism",
+				"Pentagrammic Deltohedron",
+				"Dihedral (D[2/5])",
+				"",
+				"",
+				0, 0},
+							 /* (5/3 2 2) (D3/5) */
+
+  /*  8 */	{"|2 2 5/3",	"Pentagrammic Crossed Antiprism",
+				"Pentagrammic Concave Deltohedron",
+				"Dihedral (D[3/5])",
+				"",
+				"",
+				0, 0},
+
+  /****************************************************************************
+   *		Tetrahedral
+   ***************************************************************************/
+
+							     /* (2 3 3) (T1) */
+  /*  10 */	{"3|2 3",	"Tetrahedron",
+				"Tetrahedron",
+				"Tetrahedral (T[1])",
+				"Platonic Solid",
+				"Platonic Solid",
+				15, 1},
+
+  /*  12 */	{"2 3|3",	"Truncated Tetrahedron",
+				"Triakistetrahedron",
+				"Tetrahedral (T[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				16, 6},
+							   /* (3/2 3 3) (T2) */
+  /*  14 */	{"3/2 3|3",	"Octahemioctahedron",
+				"Octahemioctacron",
+				"Tetrahedral (T[2])",
+				"",
+				"",
+				37, 68},
+
+							   /* (3/2 2 3) (T3) */
+  /*  16 */	{"3/2 3|2",	"Tetrahemihexahedron",
+				"Tetrahemihexacron",
+				"Tetrahedral (T[3])",
+				"",
+				"",
+				36, 67},
+
+  /****************************************************************************
+   *		Octahedral
+   ***************************************************************************/
+
+							     /* (2 3 4) (O1) */
+  /* 18 */	{"4|2 3",	"Octahedron",
+				"Cube",
+				"Octahedral (O[1])",
+				"Platonic Solid",
+				"Platonic Solid",
+				17, 2},
+
+  /* 20 */	{"3|2 4",	"Cube",
+				"Octahedron",
+				"Octahedral (O[1])",
+				"Platonic Solid",
+				"Platonic Solid",
+				18, 3},
+
+  /* 22 */	{"2|3 4",	"Cuboctahedron",
+				"Rhombic Dodecahedron",
+				"Octahedral (O[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				19, 11},
+
+  /* 24 */	{"2 4|3",	"Truncated Octahedron",
+				"Tetrakishexahedron",
+				"Octahedral (O[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				20, 7},
+
+  /* 26 */	{"2 3|4",	"Truncated Cube",
+				"Triakisoctahedron",
+				"Octahedral (O[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				21, 8},
+
+  /* 28 */	{"3 4|2",	"Rhombicuboctahedron",
+				"Deltoidal Icositetrahedron",
+				"Octahedral (O[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				22, 13},
+
+  /* 30 */	{"2 3 4|",	"Truncated Cuboctahedron",
+				"Disdyakisdodecahedron",
+				"Octahedral (O[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				23, 15},
+
+  /* 32, 33, 66, and 67 are chiral, existing in both left and right handed
+     (enantiomeric) forms, so it would make sense to display both versions.
+  */
+
+  /* 32 */	{"|2 3 4",	"Snub Cube",
+				"Pentagonal Icositetrahedron",
+				"Octahedral (O[1]), Chiral",
+				"Archimedean Solid",
+				"Catalan Solid",
+				24, 17},
+							  /* (3/2 4 4) (O2b) */
+
+  /* 34 */	{"3/2 4|4",	"Small Cubicuboctahedron",
+				"Small Hexacronic Icositetrahedron",
+				"Octahedral (O[2b])",
+				"",
+				"",
+				38, 69},
+							   /* (4/3 3 4) (O4) */
+
+  /* 36 */	{"3 4|4/3",	"Great Cubicuboctahedron",
+				"Great Hexacronic Icositetrahedron",
+				"Octahedral (O[4])",
+				"",
+				"",
+				50, 77},
+
+  /* 38 */	{"4/3 4|3",	"Cubohemioctahedron",
+				"Hexahemioctacron",
+				"Octahedral (O[4])",
+				"",
+				"",
+				51, 78},
+
+  /* 40 */	{"4/3 3 4|",	"Cubitruncated Cuboctahedron",
+				"Tetradyakishexahedron",
+				"Octahedral (O[4])",
+				"",
+				"",
+				52, 79},
+							   /* (3/2 2 4) (O5) */
+
+  /* 42 */	{"3/2 4|2",	"Great Rhombicuboctahedron",
+				"Great Deltoidal Icositetrahedron",
+				"Octahedral (O[5])",
+				"",
+				"",
+				59, 85},
+
+  /* 44 */	{"3/2 2 4|",	"Small Rhombihexahedron",
+				"Small Rhombihexacron",
+				"Octahedral (O[5])",
+				"",
+				"",
+				60, 86},
+							   /* (4/3 2 3) (O7) */
+
+  /* 46 */	{"2 3|4/3",	"Stellated Truncated Hexahedron",
+				"Great Triakisoctahedron",
+				"Octahedral (O[7])",
+				"",
+				"",
+				66, 92},
+
+  /* 48 */	{"4/3 2 3|",	"Great Truncated Cuboctahedron",
+				"Great Disdyakisdodecahedron",
+				"Octahedral (O[7])",
+				"",
+				"",
+				67, 93},
+							/* (4/3 3/2 2) (O11) */
+
+  /* 50 */	{"4/3 3/2 2|",	"Great Rhombihexahedron",
+				"Great Rhombihexacron",
+				"Octahedral (O[11])",
+				"",
+				"",
+				82, 103},
+
+  /****************************************************************************
+   *		Icosahedral
+   ***************************************************************************/
+
+							     /* (2 3 5) (I1) */
+  /* 52 */	{"5|2 3",	"Icosahedron",
+				"Dodecahedron",
+				"Icosahedral (I[1])",
+				"Platonic Solid",
+				"Platonic Solid",
+				25, 4},
+
+  /* 54 */	{"3|2 5",	"Dodecahedron",
+				"Icosahedron",
+				"Icosahedral (I[1])",
+				"Platonic Solid",
+				"Platonic Solid",
+				26, 5},
+
+  /* 56 */	{"2|3 5",	"Icosidodecahedron",
+				"Rhombic Triacontahedron",
+				"Icosahedral (I[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				28, 12},
+
+  /* 58 */	{"2 5|3",	"Truncated Icosahedron",
+				"Pentakisdodecahedron",
+				"Icosahedral (I[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				27, 9},
+
+  /* 60 */	{"2 3|5",	"Truncated Dodecahedron",
+				"Triakisicosahedron",
+				"Icosahedral (I[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				29, 10},
+
+  /* 62 */	{"3 5|2",	"Rhombicosidodecahedron",
+				"Deltoidal Hexecontahedron",
+				"Icosahedral (I[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				30, 14},
+
+  /* 64 */	{"2 3 5|",	"Truncated Icosidodecahedron",
+				"Disdyakistriacontahedron",
+				"Icosahedral (I[1])",
+				"Archimedean Solid",
+				"Catalan Solid",
+				31, 16},
+
+  /* 32, 33, 66, and 67 are chiral, existing in both left and right handed
+     (enantiomeric) forms, so it would make sense to display both versions.
+  */
+
+  /* 66 */	{"|2 3 5",	"Snub Dodecahedron",
+				"Pentagonal Hexecontahedron",
+				"Icosahedral (I[1]), Chiral",
+				"Archimedean Solid",
+				"Catalan Solid",
+				32, 18},
+							  /* (5/2 3 3) (I2a) */
+
+  /* 68 */	{"3|5/2 3",	"Small Ditrigonal Icosidodecahedron",
+				"Small Triambic Icosahedron",
+				"Icosahedral (I[2a])",
+				"",
+				"",
+				39, 70},
+
+  /* 70 */	{"5/2 3|3",	"Small Icosicosidodecahedron",
+				"Small Icosacronic Hexecontahedron",
+				"Icosahedral (I[2a])",
+				"",
+				"",
+				40, 71},
+
+  /* 72 */	{"|5/2 3 3",	"Small Snub Icosicosidodecahedron",
+				"Small Hexagonal Hexecontahedron",
+				"Icosahedral (I[2a])",
+				"",
+				"",
+				41, 110},
+							  /* (3/2 5 5) (I2b) */
+
+  /* 74 */	{"3/2 5|5",	"Small Dodecicosidodecahedron",
+				"Small Dodecacronic Hexecontahedron",
+				"Icosahedral (I[2b])",
+				"",
+				"",
+				42, 72},
+							   /* (2 5/2 5) (I3) */
+
+  /* 76 */	{"5|2 5/2",	"Small Stellated Dodecahedron",
+				"Great Dodecahedron",
+				"Icosahedral (I[3])",
+				"Kepler-Poinsot Solid",
+				"Kepler-Poinsot Solid",
+				43, 20},
+
+  /* 78 */	{"5/2|2 5",	"Great Dodecahedron",
+				"Small Stellated Dodecahedron",
+				"Icosahedral (I[3])",
+				"Kepler-Poinsot Solid",
+				"Kepler-Poinsot Solid",
+				44, 21},
+
+  /* 80 */	{"2|5/2 5",	"Great Dodecadodecahedron",
+				"Medial Rhombic Triacontahedron",
+				"Icosahedral (I[3])",
+				"",
+				"",
+				45, 73},
+
+  /* 82 */	{"2 5/2|5",	"Truncated Great Dodecahedron",
+				"Small Stellapentakisdodecahedron",
+				"Icosahedral (I[3])",
+				"",
+				"",
+				47, 75},
+
+  /* 84 */	{"5/2 5|2",	"Rhombidodecadodecahedron",
+				"Medial Deltoidal Hexecontahedron",
+				"Icosahedral (I[3])",
+				"",
+				"",
+				48, 76},
+
+  /* 86 */	{"2 5/2 5|",	"Small Rhombidodecahedron",
+				"Small Rhombidodecacron",
+				"Icosahedral (I[3])",
+				"",
+				"",
+				46, 74},
+
+  /* 88 */	{"|2 5/2 5",	"Snub Dodecadodecahedron",
+				"Medial Pentagonal Hexecontahedron",
+				"Icosahedral (I[3])",
+				"",
+				"",
+				49, 111},
+							   /* (5/3 3 5) (I4) */
+
+  /* 90 */	{"3|5/3 5",	"Ditrigonal Dodecadodecahedron",
+				"Medial Triambic Icosahedron",
+				"Icosahedral (I[4])",
+				"",
+				"",
+				53, 80},
+
+  /* 92 */	{"3 5|5/3",	"Great Ditrigonal Dodecicosidodecahedron",
+			       "Great Ditrigonal Dodecacronic Hexecontahedron",
+				"Icosahedral (I[4])",
+				"",
+				"",
+				54, 81},
+
+  /* 94 */	{"5/3 3|5",	"Small Ditrigonal Dodecicosidodecahedron",
+			       "Small Ditrigonal Dodecacronic Hexecontahedron",
+				"Icosahedral (I[4])",
+				"",
+				"",
+				55, 82},
+
+  /* 96 */	{"5/3 5|3",	"Icosidodecadodecahedron",
+				"Medial Icosacronic Hexecontahedron",
+				"Icosahedral (I[4])",
+				"",
+				"",
+				56, 83},
+
+  /* 98 */	{"5/3 3 5|",	"Icositruncated Dodecadodecahedron",
+				"Tridyakisicosahedron",
+				"Icosahedral (I[4])",
+				"",
+				"",
+				57, 84},
+
+  /* 100 */	{"|5/3 3 5",	"Snub Icosidodecadodecahedron",
+				"Medial Hexagonal Hexecontahedron",
+				"Icosahedral (I[4])",
+				"",
+				"",
+				58, 112},
+							  /* (3/2 3 5) (I6b) */
+
+  /* 102 */	{"3/2|3 5",	"Great Ditrigonal Icosidodecahedron",
+				"Great Triambic Icosahedron",
+				"Icosahedral (I[6b])",
+				"",
+				"",
+				61, 87},
+
+  /* 104 */	{"3/2 5|3",	"Great Icosicosidodecahedron",
+				"Great Icosacronic Hexecontahedron",
+				"Icosahedral (I[6b])",
+				"",
+				"",
+				62, 88},
+
+  /* 106 */	{"3/2 3|5",	"Small Icosihemidodecahedron",
+				"Small Icosihemidodecacron",
+				"Icosahedral (I[6b])",
+				"",
+				"",
+				63, 89},
+
+  /* 108 */	{"3/2 3 5|",	"Small Dodecicosahedron",
+				"Small Dodecicosacron",
+				"Icosahedral (I[6b])",
+				"",
+				"",
+				64, 90},
+							  /* (5/4 5 5) (I6c) */
+
+  /* 110 */	{"5/4 5|5",	"Small Dodecahemidodecahedron",
+				"Small Dodecahemidodecacron",
+				"Icosahedral (I[6c])",
+				"",
+				"",
+				65, 91},
+							   /* (2 5/2 3) (I7) */
+
+  /* 112 */	{"3|2 5/2",	"Great Stellated Dodecahedron",
+				"Great Icosahedron",
+				"Icosahedral (I[7])",
+				"Kepler-Poinsot Solid",
+				"Kepler-Poinsot Solid",
+				68, 22},
+
+  /* 114 */	{"5/2|2 3",	"Great Icosahedron",
+				"Great Stellated Dodecahedron",
+				"Icosahedral (I[7])",
+				"Kepler-Poinsot Solid",
+				"Kepler-Poinsot Solid",
+				69, 41},
+
+  /* 116 */	{"2|5/2 3",	"Great Icosidodecahedron",
+				"Great Rhombic Triacontahedron",
+				"Icosahedral (I[7])",
+				"",
+				"",
+				70, 94},
+
+  /* 118 */	{"2 5/2|3",	"Great Truncated Icosahedron",
+				"Great Stellapentakisdodecahedron",
+				"Icosahedral (I[7])",
+				"",
+				"",
+				71, 95},
+
+  /* 120 */	{"2 5/2 3|",	"Rhombicosahedron",
+				"Rhombicosacron",
+				"Icosahedral (I[7])",
+				"",
+				"",
+				72, 96},
+
+  /* 122 */	{"|2 5/2 3",	"Great Snub Icosidodecahedron",
+				"Great Pentagonal Hexecontahedron",
+				"Icosahedral (I[7])",
+				"",
+				"",
+				73, 113},
+							   /* (5/3 2 5) (I9) */
+
+  /* 124 */	{"2 5|5/3",	"Small Stellated Truncated Dodecahedron",
+				"Great Pentakisdodecahedron",
+				"Icosahedral (I[9])",
+				"",
+				"",
+				74, 97},
+
+  /* 126 */	{"5/3 2 5|",	"Truncated Dodecadodecahedron",
+				"Medial Disdyakistriacontahedron",
+				"Icosahedral (I[9])",
+				"",
+				"",
+				75, 98},
+
+  /* 128 */	{"|5/3 2 5",	"Inverted Snub Dodecadodecahedron",
+				"Medial Inverted Pentagonal Hexecontahedron",
+				"Icosahedral (I[9])",
+				"",
+				"",
+				76, 114},
+						       /* (5/3 5/2 3) (I10a) */
+
+  /* 130 */	{"5/2 3|5/3",	"Great Dodecicosidodecahedron",
+				"Great Dodecacronic Hexecontahedron",
+				"Icosahedral (I[10a])",
+				"",
+				"",
+				77, 99},
+
+  /* 132 */	{"5/3 5/2|3",	"Small Dodecahemicosahedron",
+				"Small Dodecahemicosacron",
+				"Icosahedral (I[10a])",
+				"",
+				"",
+				78, 100},
+
+  /* 134 */	{"5/3 5/2 3|",	"Great Dodecicosahedron",
+				"Great Dodecicosacron",
+				"Icosahedral (I[10a])",
+				"",
+				"",
+				79, 101},
+
+  /* 136 */	{"|5/3 5/2 3",	"Great Snub Dodecicosidodecahedron",
+				"Great Hexagonal Hexecontahedron",
+				"Icosahedral (I[10a])",
+				"",
+				"",
+				80, 115},
+							 /* (5/4 3 5) (I10b) */
+
+  /* 138 */	{"5/4 5|3",	"Great Dodecahemicosahedron",
+				"Great Dodecahemicosacron",
+				"Icosahedral (I[10b])",
+				"",
+				"",
+				81, 102},
+							  /* (5/3 2 3) (I13) */
+
+  /* 140 */	{"2 3|5/3",	"Great Stellated Truncated Dodecahedron",
+				"Great Triakisicosahedron",
+				"Icosahedral (I[13])",
+				"",
+				"",
+				83, 104},
+
+  /* 142 */	{"5/3 3|2",	"Great Rhombicosidodecahedron",
+				"Great Deltoidal Hexecontahedron",
+				"Icosahedral (I[13])",
+				"",
+				"",
+				84, 105},
+
+  /* 144 */	{"5/3 2 3|",	"Great Truncated Icosidodecahedron",
+				"Great Disdyakistriacontahedron",
+				"Icosahedral (I[13])",
+				"",
+				"",
+				87, 108},
+
+  /* 146 */	{"|5/3 2 3",	"Great Inverted Snub Icosidodecahedron",
+				"Great Inverted Pentagonal Hexecontahedron",
+				"Icosahedral (I[13])",
+				"",
+				"",
+				88, 116},
+						     /* (5/3 5/3 5/2) (I18a) */
+
+  /* 148 */	{"5/3 5/2|5/3",	"Great Dodecahemidodecahedron",
+				"Great Dodecahemidodecacron",
+				"Icosahedral (I[18a])",
+				"",
+				"",
+				86, 107},
+						       /* (3/2 5/3 3) (I18b) */
+
+  /* 150 */	{"3/2 3|5/3",	"Great Icosihemidodecahedron",
+				"Great Icosihemidodecacron",
+				"Icosahedral (I[18b])",
+				"",
+				"",
+				85, 106},
+						      /* (3/2 3/2 5/3) (I22) */
+
+  /* 152 */	{"|3/2 3/2 5/2","Small Retrosnub Icosicosidodecahedron",
+				"Small Hexagrammic Hexecontahedron",
+				"Icosahedral (I[22])",
+				"",
+				"",
+				91, 118},
+							/* (3/2 5/3 2) (I23) */
+
+  /* 154 */	{"3/2 5/3 2|",	"Great Rhombidodecahedron",
+				"Great Rhombidodecacron",
+				"Icosahedral (I[23])",
+				"",
+				"",
+				89, 109},
+
+  /* 156 */	{"|3/2 5/3 2",	"Great Retrosnub Icosidodecahedron",
+				"Great Pentagrammic Hexecontahedron",
+				"Icosahedral (I[23])",
+				"",
+				"",
+				90, 117},
+
+  /****************************************************************************
+   *		Last But Not Least
+   ***************************************************************************/
+
+  /* 158 */    {"3/2 5/3 3 5/2",	"Great Dirhombicosidodecahedron",
+				"Great Dirhombicosidodecacron",
+				"Non-Wythoffian",
+				"",
+				"",
+				92, 119}
+};
+
+static int last_uniform = sizeof (uniform) / sizeof (uniform[0]);
+
+
+
+static int unpacksym(const char *sym, Polyhedron *P);
+static int moebius(Polyhedron *P);
+static int decompose(Polyhedron *P);
+static int guessname(Polyhedron *P);
+static int newton(Polyhedron *P, int need_approx);
+static int exceptions(Polyhedron *P);
+static int count(Polyhedron *P);
+static int configuration(Polyhedron *P);
+static int vertices(Polyhedron *P);
+static int faces(Polyhedron *P);
+static int edgelist(Polyhedron *P);
+
+static Polyhedron *
+kaleido(const char *sym,
+        int need_coordinates, int need_edgelist, int need_approx,
+	int just_list)
+{
+  Polyhedron *P;
+  /*
+   * Allocate a Polyhedron structure P.
+   */
+  if (!(P = polyalloc()))
+    return 0;
+  /*
+   * Unpack input symbol into P.
+   */
+  if (!unpacksym(sym, P))
+    return 0;
+  /*
+   * Find Mebius triangle, its density and Euler characteristic.
+   */
+  if (!moebius(P))
+    return 0;
+  /*
+   * Decompose Schwarz triangle.
+   */
+  if (!decompose(P))
+    return 0;
+  /*
+   * Find the names of the polyhedron and its dual.
+   */
+  if (!guessname(P))
+    return 0;
+  if (just_list)
+    return P;
+  /*
+   * Solve Fundamental triangles, optionally printing approximations.
+   */
+  if (!newton(P,need_approx))
+    return 0;
+  /*
+   * Deal with exceptional polyhedra.
+   */
+  if (!exceptions(P))
+    return 0;
+  /*
+   * Count edges and faces, update density and characteristic if needed.
+   */
+  if (!count(P))
+    return 0;
+  /*
+   * Generate printable vertex configuration.
+   */
+  if (!configuration(P))
+    return 0;
+  /*
+   * Compute coordinates.
+   */
+  if (!need_coordinates && !need_edgelist)
+    return P;
+  if (!vertices(P))
+    return 0;
+  if (!faces (P))
+    return 0;
+  /*
+   * Compute edgelist.
+   */
+  if (!need_edgelist)
+    return P;
+  if (!edgelist(P))
+    return 0;
+  return P;
+}
+
+/*
+ * Allocate a blank Polyhedron structure and initialize some of its nonblank
+ * fields.
+ *
+ * Array and matrix field are allocated when needed.
+ */
+static Polyhedron *
+polyalloc()
+{
+  Polyhedron *P;
+  Calloc(P, 1, Polyhedron);
+  P->index = -1;
+  P->even = -1;
+  P->K = 2;
+  return P;
+}
+
+/*
+ * Free the struture allocated by polyalloc(), as well as all the array and
+ * matrix fields.
+ */
+static void
+polyfree(Polyhedron *P)
+{
+  Free(P->Fi);
+  Free(P->n);
+  Free(P->m);
+  Free(P->gamma);
+  Free(P->rot);
+  Free(P->snub);
+  Free(P->firstrot);
+  Free(P->anti);
+  Free(P->ftype);
+  Free(P->polyform);
+  Free(P->config);
+  if (P->index < 0) {
+    Free(P->name);
+    Free(P->dual_name);
+  }
+  Free(P->v);
+  Free(P->f);
+  Matfree(P->e, 2);
+  Matfree(P->dual_e, 2);
+  Matfree(P->incid, P->M);
+  Matfree(P->adj, P->M);
+  free(P);
+}
+
+static void *
+matalloc(int rows, int row_size)
+{
+  void **mat;
+  int i = 0;
+  if (!(mat = malloc(rows * sizeof (void *))))
+    return 0;
+  while ((mat[i] = malloc(row_size)) && ++i < rows)
+    ;
+  if (i == rows)
+    return (void *)mat;
+  while (--i >= 0)
+    free(mat[i]);
+  free(mat);
+  return 0;
+}
+
+static void
+matfree(void *mat, int rows)
+{
+  while (--rows >= 0)
+    free(((void **)mat)[rows]);
+  free(mat);
+}
+
+/*
+ * compute the mathematical modulus function.
+ */
+static int
+mod (int i, int j)
+{
+  return (i%=j)>=0?i:j<0?i-j:i+j;
+}
+
+
+/*
+ * Find the numerator and the denominator using the Euclidean algorithm.
+ */
+static void
+frac(double x)
+{
+  static const Fraction zero = {0,1}, inf = {1,0};
+  Fraction r0, r;
+  long f;
+  double s = x;
+  r = zero;
+  frax = inf;
+  for (;;) {
+    if (fabs(s) > (double) MAXLONG)
+      return;
+    f = (long) floor (s);
+    r0 = r;
+    r = frax;
+    frax.n = frax.n * f + r0.n;
+    frax.d = frax.d * f + r0.d;
+    if (x == (double)frax.n/(double)frax.d)
+      return;
+    s = 1 / (s - f);
+  }
+}
+
+
+/*
+ * Unpack input symbol: Wythoff symbol or an index to uniform[]. The symbol is
+ * a # followed by a number, or a three fractions and a bar in some order. We
+ * allow no bars only if it result from the input symbol #80.
+ */
+static int
+unpacksym(const char *sym, Polyhedron *P)
+{
+  int i = 0, n, d, bars = 0;
+  char c;
+  while ((c = *sym++) && isspace(c))
+    ;
+  if (!c) Err("no data");
+  if (c == '#') {
+    while ((c = *sym++) && isspace(c))
+      ;
+    if (!c)
+      Err("no digit after #");
+    if (!isdigit(c))
+      Err("not a digit");
+    n = c - '0';
+    while ((c = *sym++) && isdigit(c))
+      n = n * 10 + c - '0';
+    if (!n)
+      Err("zero index");
+    if (n > last_uniform)
+      Err("index too big");
+    sym--;
+    while ((c = *sym++) && isspace(c))
+      ;
+    if (c)
+      Err("data exceeded");
+    sym = uniform[P->index = n - 1].Wythoff;
+  } else
+    sym--;
+
+  for (;;) {
+    while ((c = *sym++) && isspace(c))
+      ;
+    if (!c) {
+      if (i == 4 && (bars || P->index == last_uniform - 1))
+        return 1;
+      if (!bars)
+        Err("no bars");
+      Err("not enough fractions");
+    }
+    if (i == 4)
+      Err("data exceeded");
+    if (c == '|'){
+      if (++bars > 1)
+        Err("too many bars");
+      P->p[i++] = 0;
+      continue;
+    }
+    if (!isdigit(c))
+      Err("not a digit");
+    n = c - '0';
+    while ((c = *sym++) && isdigit(c))
+      n = n * 10 + c - '0';
+    if (c && isspace (c))
+      while ((c = *sym++) && isspace(c))
+        ;
+    if (c != '/') {
+      sym--;
+      if ((P->p[i++] = n) <= 1)
+        Err("fraction<=1");
+      continue;
+    }
+    while ((c = *sym++) && isspace(c))
+      ;
+    if (!c || !isdigit(c))
+      return 0;
+    d = c - '0';
+    while ((c = *sym++) && isdigit(c))
+      d = d * 10 + c - '0';
+    if (!d)
+      Err("zero denominator");
+    sym--;
+    if ((P->p[i++] = (double) n / d) <= 1)
+      Err("fraction<=1");
+  }
+}
+
+/*
+ * Using Wythoff symbol (p|qr, pq|r, pqr| or |pqr), find the Moebius triangle
+ * (2 3 K) (or (2 2 n)) of the Schwarz triangle (pqr), the order g of its
+ * symmetry group, its Euler characteristic chi, and its covering density D.
+ * g is the number of copies of (2 3 K) covering the sphere, i.e.,
+ *
+ *		g * pi * (1/2 + 1/3 + 1/K - 1) = 4 * pi
+ *
+ * D is the number of times g copies of (pqr) cover the sphere, i.e.
+ *
+ *		D * 4 * pi = g * pi * (1/p + 1/q + 1/r - 1)
+ *
+ * chi is V - E + F, where F = g is the number of triangles, E = 3*g/2 is the
+ * number of triangle edges, and V = Vp+ Vq+ Vr, with Vp = g/(2*np) being the
+ * number of vertices with angle pi/p (np is the numerator of p).
+ */
+static int
+moebius(Polyhedron *P)
+{
+  int twos = 0, j, len = 1;
+  /*
+   * Arrange Wythoff symbol in a presentable form. In the same time check the
+   * restrictions on the three fractions: They all have to be greater then one,
+   * and the numerators 4 or 5 cannot occur together.  We count the ocurrences
+   * of 2 in `two', and save the largest numerator in `P->K', since they
+   * reflect on the symmetry group.
+   */
+  P->K = 2;
+  if (P->index == last_uniform - 1) {
+    Malloc(P->polyform, ++len, char);
+    strcpy(P->polyform, "|");
+  } else
+    Calloc(P->polyform, len, char);
+  for (j = 0; j < 4; j++) {
+    if (P->p[j]) {
+      char *s;
+      Sprintfrac(s, P->p[j]);
+      if (j && P->p[j-1]) {
+        Realloc(P->polyform, len += strlen (s) + 1, char);
+        strcat(P->polyform, " ");
+      } else
+        Realloc (P->polyform, len += strlen (s), char);
+      strcat(P->polyform, s);
+      free(s);
+      if (P->p[j] != 2) {
+        int k;
+        if ((k = numerator (P->p[j])) > P->K) {
+          if (P->K == 4)
+            break;
+          P->K = k;
+        } else if (k < P->K && k == 4)
+          break;
+      } else
+        twos++;
+    } else  {
+      Realloc(P->polyform, ++len, char);
+      strcat(P->polyform, "|");
+    }
+  }
+  /*
+   * Find the symmetry group P->K (where 2, 3, 4, 5 represent the dihedral,
+   * tetrahedral, octahedral and icosahedral groups, respectively), and its
+   * order P->g.
+   */
+  if (twos >= 2) {/* dihedral */
+    P->g = 4 * P->K;
+    P->K = 2;
+  } else {
+    if (P->K > 5)
+      Err("numerator too large");
+    P->g = 24 * P->K / (6 - P->K);
+  }
+  /*
+   * Compute the nominal density P->D and Euler characteristic P->chi.
+   * In few exceptional cases, these values will be modified later.
+   */
+  if (P->index != last_uniform - 1) {
+    int i;
+    P->D = P->chi = - P->g;
+    for (j = 0; j < 4; j++) if (P->p[j]) {
+      P->chi += i = P->g / numerator(P->p[j]);
+      P->D += i * denominator(P->p[j]);
+    }
+    P->chi /= 2;
+    P->D /= 4;
+    if (P->D <= 0)
+      Err("nonpositive density");
+  }
+  return 1;
+}
+
+/*
+ * Decompose Schwarz triangle into N right triangles and compute the vertex
+ * count V and the vertex valency M.  V is computed from the number g of
+ * Schwarz triangles in the cover, divided by the number of triangles which
+ * share a vertex. It is halved for one-sided polyhedra, because the
+ * kaleidoscopic construction really produces a double orientable covering of
+ * such polyhedra. All q' q|r are of the "hemi" type, i.e. have equatorial {2r}
+ * faces, and therefore are (except 3/2 3|3 and the dihedra 2 2|r) one-sided. A
+ * well known example is 3/2 3|4, the "one-sided heptahedron". Also, all p q r|
+ * with one even denominator have a crossed parallelogram as a vertex figure,
+ * and thus are one-sided as well.
+ */
+static int
+decompose(Polyhedron *P)
+{
+  int j, J, *s, *t;
+  if (!P->p[1]) { /* p|q r */
+    P->N = 2;
+    P->M = 2 * numerator(P->p[0]);
+    P->V = P->g / P->M;
+    Malloc(P->n, P->N, double);
+    Malloc(P->m, P->N, double);
+    Malloc(P->rot, P->M, int);
+    s = P->rot;
+    for (j = 0; j < 2; j++) {
+      P->n[j] = P->p[j+2];
+      P->m[j] = P->p[0];
+    }
+    for (j = P->M / 2; j--;) {
+      *s++ = 0;
+      *s++ = 1;
+    }
+  } else if (!P->p[2]) { /* p q|r */
+    P->N = 3;
+    P->M = 4;
+    P->V = P->g / 2;
+    Malloc(P->n, P->N, double);
+    Malloc(P->m, P->N, double);
+    Malloc(P->rot, P->M, int);
+    s = P->rot;
+    P->n[0] = 2 * P->p[3];
+    P->m[0] = 2;
+    for (j = 1; j < 3; j++) {
+      P->n[j] = P->p[j-1];
+      P->m[j] = 1;
+      *s++ = 0;
+      *s++ = j;
+    }
+    if (fabs(P->p[0] - compl (P->p[1])) < DBL_EPSILON) {/* p = q' */
+      /* P->p[0]==compl(P->p[1]) should work.  However, MSDOS
+       * yeilds a 7e-17 difference! Reported by Jim Buddenhagen
+       * <jb1556@daditz.sbc.com> */
+      P->hemi = 1;
+      P->D = 0;
+      if (P->p[0] != 2 && !(P->p[3] == 3 && (P->p[0] == 3 ||
+                                             P->p[1] == 3))) {
+        P->onesided = 1;
+        P->V /= 2;
+        P->chi /= 2;
+      }
+    }
+  } else if (!P->p[3]) { /* p q r| */
+    P->M = P->N = 3;
+    P->V = P->g;
+    Malloc(P->n, P->N, double);
+    Malloc(P->m, P->N, double);
+    Malloc(P->rot, P->M, int);
+    s = P->rot;
+    for (j = 0; j < 3; j++) {
+      if (!(denominator(P->p[j]) % 2)) {
+        /* what happens if there is more then one even denominator? */
+        if (P->p[(j+1)%3] != P->p[(j+2)%3]) { /* needs postprocessing */
+          P->even = j;/* memorize the removed face */
+          P->chi -= P->g / numerator(P->p[j]) / 2;
+          P->onesided = 1;
+          P->D = 0;
+        } else {/* for p = q we get a double 2 2r|p */
+		/* noted by Roman Maeder <maeder@inf.ethz.ch> for 4 4 3/2| */
+		/* Euler characteristic is still wrong */
+          P->D /= 2;
+        }
+        P->V /= 2;
+      }
+      P->n[j] = 2 * P->p[j];
+      P->m[j] = 1;
+      *s++ = j;
+    }
+  } else { /* |p q r - snub polyhedron */
+    P->N = 4;
+    P->M = 6;
+    P->V = P->g / 2;/* Only "white" triangles carry a vertex */
+    Malloc(P->n, P->N, double);
+    Malloc(P->m, P->N, double);
+    Malloc(P->rot, P->M, int);
+    Malloc(P->snub, P->M, int);
+    s = P->rot;
+    t = P->snub;
+    P->m[0] = P->n[0] = 3;
+    for (j = 1; j < 4; j++) {
+      P->n[j] = P->p[j];
+      P->m[j] = 1;
+      *s++ = 0;
+      *s++ = j;
+      *t++ = 1;
+      *t++ = 0;
+    }
+  }
+  /*
+   * Sort the fundamental triangles (using bubble sort) according to decreasing
+   * n[i], while pushing the trivial triangles (n[i] = 2) to the end.
+   */
+  J = P->N - 1;
+  while (J) {
+    int last;
+    last = J;
+    J = 0;
+    for (j = 0; j < last; j++) {
+      if ((P->n[j] < P->n[j+1] || P->n[j] == 2) && P->n[j+1] != 2) {
+        int i;
+        double temp;
+        temp = P->n[j];
+        P->n[j] = P->n[j+1];
+        P->n[j+1] = temp;
+        temp = P->m[j];
+        P->m[j] = P->m[j+1];
+        P->m[j+1] = temp;
+        for (i = 0; i < P->M; i++) {
+          if (P->rot[i] == j)
+            P->rot[i] = j+1;
+          else if (P->rot[i] == j+1)
+            P->rot[i] = j;
+        }
+        if (P->even != -1) {
+          if (P->even == j)
+            P->even = j+1;
+          else if (P->even == j+1)
+            P->even = j;
+        }
+        J = j;
+      }
+    }
+  }
+  /*
+   *  Get rid of repeated triangles.
+   */
+  for (J = 0; J < P->N && P->n[J] != 2;J++) {
+    int k, i;
+    for (j = J+1; j < P->N && P->n[j]==P->n[J]; j++)
+      P->m[J] += P->m[j];
+    k = j - J - 1;
+    if (k) {
+      for (i = j; i < P->N; i++) {
+        P->n[i - k] = P->n[i];
+        P->m[i - k] = P->m[i];
+      }
+      P->N -= k;
+      for (i = 0; i < P->M; i++) {
+        if (P->rot[i] >= j)
+          P->rot[i] -= k;
+        else if (P->rot[i] > J)
+          P->rot[i] = J;
+      }
+      if (P->even >= j)
+        P->even -= k;
+    }
+  }
+  /*
+   * Get rid of trivial triangles.
+   */
+  if (!J)
+    J = 1; /* hosohedron */
+  if (J < P->N) {
+    int i;
+    P->N = J;
+    for (i = 0; i < P->M; i++) {
+      if (P->rot[i] >= P->N) {
+        for (j = i + 1; j < P->M; j++) {
+          P->rot[j-1] = P->rot[j];
+          if (P->snub)
+            P->snub[j-1] = P->snub[j];
+        }
+        P->M--;
+      }
+    }
+  }
+  /*
+   * Truncate arrays
+   */
+  Realloc(P->n, P->N, double);
+  Realloc(P->m, P->N, double);
+  Realloc(P->rot, P->M, int);
+  if (P->snub)
+    Realloc(P->snub, P->M, int);
+  return 1;
+}
+
+
+static int dihedral(Polyhedron *P, const char *name, const char *dual_name);
+
+
+/*
+ * Get the polyhedron name, using standard list or guesswork. Ideally, we
+ * should try to locate the Wythoff symbol in the standard list (unless, of
+ * course, it is dihedral), after doing few normalizations, such as sorting
+ * angles and splitting isoceles triangles.
+ */
+static int
+guessname(Polyhedron *P)
+{
+  if (P->index != -1) {/* tabulated */
+    P->name = uniform[P->index].name;
+    P->dual_name = uniform[P->index].dual;
+    P->group = uniform[P->index].group;
+    P->class = uniform[P->index].class;
+    P->dual_class = uniform[P->index].dual_class;
+    return 1;
+  } else if (P->K == 2) {/* dihedral nontabulated */
+    if (!P->p[0]) {
+      if (P->N == 1) {
+        Malloc(P->name, sizeof ("Octahedron"), char);
+        Malloc(P->dual_name, sizeof ("Cube"), char);
+        strcpy(P->name, "Octahedron");
+        strcpy(P->dual_name, "Cube");
+        return 1;
+      }
+      P->gon = P->n[0] == 3 ? P->n[1] : P->n[0];
+      if (P->gon >= 2)
+        return dihedral(P, "Antiprism", "Deltohedron");
+      else
+        return dihedral(P, "Crossed Antiprism", "Concave Deltohedron");
+    } else if (!P->p[3] ||
+               (!P->p[2] &&
+                P->p[3] == 2)) {
+      if (P->N == 1) {
+        Malloc(P->name, sizeof("Cube"), char);
+        Malloc(P->dual_name, sizeof("Octahedron"), char);
+        strcpy(P->name, "Cube");
+        strcpy(P->dual_name, "Octahedron");
+        return 1;
+      }
+      P->gon = P->n[0] == 4 ? P->n[1] : P->n[0];
+      return dihedral(P, "Prism", "Dipyramid");
+    } else if (!P->p[1] && P->p[0] != 2) {
+      P->gon = P->m[0];
+      return dihedral(P, "Hosohedron", "Dihedron");
+    } else {
+      P->gon = P->n[0];
+      return dihedral(P, "Dihedron", "Hosohedron");
+    }
+  } else {/* other nontabulated */
+    static const char *pre[] = {"Tetr", "Oct", "Icos"};
+    Malloc(P->name, 50, char);
+    Malloc(P->dual_name, 50, char);
+    sprintf(P->name, "%sahedral ", pre[P->K - 3]);
+    if (P->onesided)
+      strcat (P->name, "One-Sided ");
+    else if (P->D == 1)
+      strcat(P->name, "Convex ");
+    else
+      strcat(P->name, "Nonconvex ");
+    strcpy(P->dual_name, P->name);
+    strcat(P->name, "Isogonal Polyhedron");
+    strcat(P->dual_name, "Isohedral Polyhedron");
+    Realloc(P->name, strlen (P->name) + 1, char);
+    Realloc(P->dual_name, strlen (P->dual_name) + 1, char);
+    return 1;
+  }
+}
+
+static int
+dihedral(Polyhedron *P, const char *name, const char *dual_name)
+{
+  char *s;
+  int i;
+  Sprintfrac(s, P->gon < 2 ? compl (P->gon) : P->gon);
+  i = strlen(s) + sizeof ("-gonal ");
+  Malloc(P->name, i + strlen (name), char);
+  Malloc(P->dual_name, i + strlen (dual_name), char);
+  sprintf(P->name, "%s-gonal %s", s, name);
+  sprintf(P->dual_name, "%s-gonal %s", s, dual_name);
+  free(s);
+  return 1;
+}
+
+/*
+ * Solve the fundamental right spherical triangles.
+ * If need_approx is set, print iterations on standard error.
+ */
+static int
+newton(Polyhedron *P, int need_approx)
+{
+  /*
+   * First, we find initial approximations.
+   */
+  int j;
+  double cosa;
+  Malloc(P->gamma, P->N, double);
+  if (P->N == 1) {
+    P->gamma[0] = M_PI / P->m[0];
+    return 1;
+  }
+  for (j = 0; j < P->N; j++)
+    P->gamma[j] = M_PI / 2 - M_PI / P->n[j];
+  errno = 0; /* may be non-zero from some reason */
+  /*
+   * Next, iteratively find closer approximations for gamma[0] and compute
+   * other gamma[j]'s from Napier's equations.
+   */
+  if (need_approx)
+    fprintf(stderr, "Solving %s\n", P->polyform);
+  for (;;) {
+    double delta = M_PI, sigma = 0;
+    for (j = 0; j < P->N; j++) {
+      if (need_approx)
+        fprintf(stderr, "%-20.15f", P->gamma[j]);
+      delta -= P->m[j] * P->gamma[j];
+    }
+    if (need_approx)
+      printf("(%g)\n", delta);
+    if (fabs(delta) < 11 * DBL_EPSILON)
+      return 1;
+    /* On a RS/6000, fabs(delta)/DBL_EPSILON may occilate between 8 and
+     * 10. Reported by David W. Sanderson <dws@ssec.wisc.edu> */
+    for (j = 0; j < P->N; j++)
+      sigma += P->m[j] * tan(P->gamma[j]);
+    P->gamma[0] += delta * tan(P->gamma[0]) / sigma;
+    if (P->gamma[0] < 0 || P->gamma[0] > M_PI)
+      Err("gamma out of bounds");
+    cosa = cos(M_PI / P->n[0]) / sin(P->gamma[0]);
+    for (j = 1; j < P->N; j++)
+      P->gamma[j] = asin(cos(M_PI / P->n[j]) / cosa);
+    if (errno)
+      Err(strerror(errno));
+  }
+}
+
+/*
+ * Postprocess pqr| where r has an even denominator (cf. Coxeter &al. Sec.9).
+ * Remove the {2r} and add a retrograde {2p} and retrograde {2q}.
+ */
+static int
+exceptions(Polyhedron *P)
+{
+  int j;
+  if (P->even != -1) {
+    P->M = P->N = 4;
+    Realloc(P->n, P->N, double);
+    Realloc(P->m, P->N, double);
+    Realloc(P->gamma, P->N, double);
+    Realloc(P->rot, P->M, int);
+    for (j = P->even + 1; j < 3; j++) {
+      P->n[j-1] = P->n[j];
+      P->gamma[j-1] = P->gamma[j];
+    }
+    P->n[2] = compl(P->n[1]);
+    P->gamma[2] = - P->gamma[1];
+    P->n[3] = compl(P->n[0]);
+    P->m[3] = 1;
+    P->gamma[3] = - P->gamma[0];
+    P->rot[0] = 0;
+    P->rot[1] = 1;
+    P->rot[2] = 3;
+    P->rot[3] = 2;
+  }
+
+  /*
+   * Postprocess the last polyhedron |3/2 5/3 3 5/2 by taking a |5/3 3 5/2,
+   * replacing the three snub triangles by four equatorial squares and adding
+   * the missing {3/2} (retrograde triangle, cf. Coxeter &al. Sec. 11).
+   */
+  if (P->index == last_uniform - 1) {
+    P->N = 5;
+    P->M = 8;
+    Realloc(P->n, P->N, double);
+    Realloc(P->m, P->N, double);
+    Realloc(P->gamma, P->N, double);
+    Realloc(P->rot, P->M, int);
+    Realloc(P->snub, P->M, int);
+    P->hemi = 1;
+    P->D = 0;
+    for (j = 3; j; j--) {
+      P->m[j] = 1;
+      P->n[j] = P->n[j-1];
+      P->gamma[j] = P->gamma[j-1];
+    }
+    P->m[0] = P->n[0] = 4;
+    P->gamma[0] = M_PI / 2;
+    P->m[4] = 1;
+    P->n[4] = compl(P->n[1]);
+    P->gamma[4] = - P->gamma[1];
+    for (j = 1; j < 6; j += 2) P->rot[j]++;
+    P->rot[6] = 0;
+    P->rot[7] = 4;
+    P->snub[6] = 1;
+    P->snub[7] = 0;
+  }
+  return 1;
+}
+
+/*
+ * Compute edge and face counts, and update D and chi.  Update D in the few
+ * cases the density of the polyhedron is meaningful but different than the
+ * density of the corresponding Schwarz triangle (cf. Coxeter &al., p. 418 and
+ * p. 425).
+ * In these cases, spherical faces of one type are concave (bigger than a
+ * hemisphere), and the actual density is the number of these faces less the
+ * computed density.  Note that if j != 0, the assignment gamma[j] = asin(...)
+ * implies gamma[j] cannot be obtuse.  Also, compute chi for the only
+ * non-Wythoffian polyhedron.
+ */
+static int
+count(Polyhedron *P)
+{
+  int j, temp;
+  Malloc(P->Fi, P->N, int);
+  for (j = 0; j < P->N; j++) {
+    P->E += temp = P->V * numerator(P->m[j]);
+    P->F += P->Fi[j] = temp / numerator(P->n[j]);
+  }
+  P->E /= 2;
+  if (P->D && P->gamma[0] > M_PI / 2)
+    P->D = P->Fi[0] - P->D;
+  if (P->index == last_uniform - 1)
+    P->chi = P->V - P->E + P->F;
+  return 1;
+}
+
+/*
+ * Generate a printable vertex configuration symbol.
+ */
+static int
+configuration(Polyhedron *P)
+{
+  int j, len = 2;
+  for (j = 0; j < P->M; j++) {
+    char *s;
+    Sprintfrac(s, P->n[P->rot[j]]);
+    len += strlen (s) + 2;
+    if (!j) {
+      Malloc(P->config, len, char);
+/*      strcpy(P->config, "(");*/
+      strcpy(P->config, "");
+    } else {
+      Realloc(P->config, len, char);
+      strcat(P->config, ", ");
+    }
+    strcat(P->config, s);
+    free(s);
+  }
+/*  strcat (P->config, ")");*/
+  if ((j = denominator (P->m[0])) != 1) {
+    char s[MAXDIGITS + 2];
+    sprintf(s, "/%d", j);
+    Realloc(P->config, len + strlen (s), char);
+    strcat(P->config, s);
+  }
+  return 1;
+}
+
+/*
+ * Compute polyhedron vertices and vertex adjecency lists.
+ * The vertices adjacent to v[i] are v[adj[0][i], v[adj[1][i], ...
+ * v[adj[M-1][i], ordered counterclockwise.  The algorith is a BFS on the
+ * vertices, in such a way that the vetices adjacent to a givem vertex are
+ * obtained from its BFS parent by a cyclic sequence of rotations. firstrot[i]
+ * points to the first  rotaion in the sequence when applied to v[i]. Note that
+ * for non-snub polyhedra, the rotations at a child are opposite in sense when
+ * compared to the rotations at the parent. Thus, we fill adj[*][i] from the
+ * end to signify clockwise rotations. The firstrot[] array is not needed for
+ * display thus it is freed after being used for face computations below.
+ */
+static int
+vertices(Polyhedron *P)
+{
+  int i, newV = 2;
+  double cosa;
+  Malloc(P->v, P->V, Vector);
+  Matalloc(P->adj, P->M, P->V, int);
+  Malloc(P->firstrot, P->V, int); /* temporary , put in Polyhedron
+                                     structure so that may be freed on
+                                     error */
+  cosa = cos(M_PI / P->n[0]) / sin(P->gamma[0]);
+  P->v[0].x = 0;
+  P->v[0].y = 0;
+  P->v[0].z = 1;
+  P->firstrot[0] = 0;
+  P->adj[0][0] = 1;
+  P->v[1].x = 2 * cosa * sqrt(1 - cosa * cosa);
+  P->v[1].y = 0;
+  P->v[1].z = 2 * cosa * cosa - 1;
+  if (!P->snub) {
+    P->firstrot[1] = 0;
+    P->adj[0][1] = -1;/* start the other side */
+    P->adj[P->M-1][1] = 0;
+  } else {
+    P->firstrot[1] = P->snub[P->M-1] ? 0 : P->M-1 ;
+    P->adj[0][1] = 0;
+  }
+  for (i = 0; i < newV; i++) {
+    int j, k;
+    int last, one, start, limit;
+    if (P->adj[0][i] == -1) {
+      one = -1; start = P->M-2; limit = -1;
+    } else {
+      one = 1; start = 1; limit = P->M;
+    }
+    k = P->firstrot[i];
+    for (j = start; j != limit; j += one) {
+      Vector temp;
+      int J;
+      temp = rotate (P->v[P->adj[j-one][i]], P->v[i],
+                     one * 2 * P->gamma[P->rot[k]]);
+      for (J=0; J<newV && !same(P->v[J],temp,BIG_EPSILON); J++)
+        ;/* noop */
+      P->adj[j][i] = J;
+      last = k;
+      if (++k == P->M)
+        k = 0;
+      if (J == newV) { /* new vertex */
+        if (newV == P->V) Err ("too many vertices");
+        P->v[newV++] = temp;
+        if (!P->snub) {
+          P->firstrot[J] = k;
+          if (one > 0) {
+            P->adj[0][J] = -1;
+            P->adj[P->M-1][J] = i;
+          } else {
+            P->adj[0][J] = i;
+          }
+        } else {
+          P->firstrot[J] = !P->snub[last] ? last :
+            !P->snub[k] ? (k+1)%P->M : k ;
+          P->adj[0][J] = i;
+        }
+      }
+    }
+  }
+  return 1;
+}
+
+/*
+ * Compute polyhedron faces (dual vertices) and incidence matrices.
+ * For orientable polyhedra, we can distinguish between the two faces meeting
+ * at a given directed edge and identify the face on the left and the face on
+ * the right, as seen from the outside.  For one-sided polyhedra, the vertex
+ * figure is a papillon (in Coxeter &al.  terminology, a crossed parallelogram)
+ * and the two faces meeting at an edge can be identified as the side face
+ * (n[1] or n[2]) and the diagonal face (n[0] or n[3]).
+ */
+static int
+faces(Polyhedron *P)
+{
+  int i, newF = 0;
+  Malloc (P->f, P->F, Vector);
+  Malloc (P->ftype, P->F, int);
+  Matalloc (P->incid, P->M, P->V, int);
+  P->minr = 1 / fabs (tan (M_PI / P->n[P->hemi]) * tan (P->gamma[P->hemi]));
+  for (i = P->M; --i>=0;) {
+    int j;
+    for (j = P->V; --j>=0;)
+      P->incid[i][j] = -1;
+  }
+  for (i = 0; i < P->V; i++) {
+    int j;
+    for (j = 0; j < P->M; j++) {
+      int i0, J;
+      int pap=0;/* papillon edge type */
+      if (P->incid[j][i] != -1)
+        continue;
+      P->incid[j][i] = newF;
+      if (newF == P->F)
+        Err("too many faces");
+      P->f[newF] = pole(P->minr, P->v[i], P->v[P->adj[j][i]],
+                        P->v[P->adj[mod(j + 1, P->M)][i]]);
+      P->ftype[newF] = P->rot[mod(P->firstrot[i] + ((P->adj[0][i] <
+                                                     P->adj[P->M - 1][i])
+                                                    ?  j
+                                                    : -j - 2),
+                                  P->M)];
+      if (P->onesided)
+        pap = (P->firstrot[i] + j) % 2;
+      i0 = i;
+      J = j;
+      for (;;) {
+        int k;
+        k = i0;
+        if ((i0 = P->adj[J][k]) == i) break;
+        for (J = 0; J < P->M && P->adj[J][i0] != k; J++)
+          ;/* noop */
+        if (J == P->M)
+          Err("too many faces");
+        if (P->onesided && (J + P->firstrot[i0]) % 2 == pap) {
+          P->incid [J][i0] = newF;
+          if (++J >= P->M)
+            J = 0;
+        } else {
+          if (--J < 0)
+            J = P->M - 1;
+          P->incid [J][i0] = newF;
+        }
+      }
+      newF++;
+    }
+  }
+  Free(P->firstrot);
+  Free(P->rot);
+  Free(P->snub);
+  return 1;
+}
+
+/*
+ * Compute edge list and graph polyhedron and dual.
+ * If the polyhedron is of the "hemi" type, each edge has one finite vertex and
+ * one ideal vertex. We make sure the latter is always the out-vertex, so that
+ * the edge becomes a ray (half-line).  Each ideal vertex is represented by a
+ * unit Vector, and the direction of the ray is either parallel or
+ * anti-parallel this Vector. We flag this in the array P->anti[E].
+ */
+static int
+edgelist(Polyhedron *P)
+{
+  int i, j, *s, *t, *u;
+  Matalloc(P->e, 2, P->E, int);
+  Matalloc(P->dual_e, 2, P->E, int);
+  s = P->e[0];
+  t = P->e[1];
+  for (i = 0; i < P->V; i++)
+    for (j = 0; j < P->M; j++)
+      if (i < P->adj[j][i]) {
+        *s++ = i;
+        *t++ = P->adj[j][i];
+      }
+  s = P->dual_e[0];
+  t = P->dual_e[1];
+  if (!P->hemi)
+    P->anti = 0;
+  else
+    Malloc(P->anti, P->E, int);
+  u = P->anti;
+  for (i = 0; i < P->V; i++)
+    for (j = 0; j < P->M; j++)
+      if (i < P->adj[j][i])
+        {
+          if (!u) {
+            *s++ = P->incid[mod(j-1,P->M)][i];
+            *t++ = P->incid[j][i];
+          } else {
+            if (P->ftype[P->incid[j][i]]) {
+              *s = P->incid[j][i];
+              *t = P->incid[mod(j-1,P->M)][i];
+            } else {
+              *s = P->incid[mod(j-1,P->M)][i];
+              *t = P->incid[j][i];
+            }
+            *u++ = dot(P->f[*s++], P->f[*t++]) > 0;
+          }
+        }
+  return 1;
+}
+
+
+static char *
+sprintfrac(double x)
+{
+  char *s;
+  frac (x);
+  if (!frax.d) {
+    Malloc(s, sizeof ("infinity"), char);
+    strcpy(s, "infinity");
+  } else if (frax.d == 1) {
+    char n[MAXDIGITS + 1];
+    sprintf(n, "%ld", frax.n);
+    Malloc(s, strlen (n) +  1, char);
+    strcpy(s, n);
+  } else {
+    char n[MAXDIGITS + 1], d[MAXDIGITS + 1];
+    sprintf(n, "%ld", frax.n);
+    sprintf(d, "%ld", frax.d);
+    Malloc(s, strlen (n) + strlen (d) + 2, char);
+    sprintf(s, "%s/%s", n, d);
+  }
+  return s;
+}
+
+static double
+dot(Vector a, Vector b)
+{
+  return a.x * b.x + a.y * b.y + a.z * b.z;
+}
+
+static Vector
+scale(double k, Vector a)
+{
+  a.x *= k;
+  a.y *= k;
+  a.z *= k;
+  return a;
+}
+
+static Vector
+diff(Vector a, Vector b)
+{
+  a.x -= b.x;
+  a.y -= b.y;
+  a.z -= b.z;
+  return a;
+}
+
+static Vector
+cross(Vector a, Vector b)
+{
+  Vector p;
+  p.x = a.y * b.z - a.z * b.y;
+  p.y = a.z * b.x - a.x * b.z;
+  p.z = a.x * b.y - a.y * b.x;
+  return p;
+}
+
+static Vector
+sum(Vector a, Vector b)
+{
+  a.x += b.x;
+  a.y += b.y;
+  a.z += b.z;
+  return a;
+}
+
+static Vector
+sum3(Vector a, Vector b, Vector c)
+{
+  a.x += b.x + c.x;
+  a.y += b.y + c.y;
+  a.z += b.z + c.z;
+  return a;
+}
+
+static Vector
+rotate(Vector vertex, Vector axis, double angle)
+{
+  Vector p;
+  p = scale(dot (axis, vertex), axis);
+  return sum3(p, scale(cos(angle), diff(vertex, p)),
+              scale(sin(angle), cross(axis, vertex)));
+}
+
+static Vector x, y, z;
+
+/*
+ * rotate the standard frame
+ */
+static void
+rotframe(double azimuth, double elevation, double angle)
+{
+  static const Vector X = {1,0,0}, Y = {0,1,0}, Z = {0,0,1};
+  Vector axis;
+
+  axis = rotate(rotate (X, Y, elevation), Z, azimuth);
+  x = rotate(X, axis, angle);
+  y = rotate(Y, axis, angle);
+  z = rotate(Z, axis, angle);
+}
+
+/*
+ * rotate an array of n Vectors
+ */
+static void
+rotarray(Vector *new, Vector *old, int n)
+{
+  while (n--) {
+    *new++ = sum3(scale(old->x, x), scale(old->y, y), scale(old->z, z));
+    old++;
+  }
+}
+
+static int
+same(Vector a, Vector b, double epsilon)
+{
+  return fabs(a.x - b.x) < epsilon && fabs(a.y - b.y) < epsilon
+    && fabs(a.z - b.z) < epsilon;
+}
+
+/*
+ * Compute the polar reciprocal of the plane containing a, b and c:
+ *
+ * If this plane does not contain the origin, return p such that
+ *	dot(p,a) = dot(p,b) = dot(p,b) = r.
+ *
+ * Otherwise, return p such that
+ *	dot(p,a) = dot(p,b) = dot(p,c) = 0
+ * and
+ *	dot(p,p) = 1.
+ */
+static Vector
+pole(double r, Vector a, Vector b, Vector c)
+{
+  Vector p;
+  double k;
+  p = cross(diff(b, a), diff(c, a));
+  k = dot(p, a);
+  if (fabs(k) < 1e-6)
+    return scale(1 / sqrt(dot(p, p)), p);
+  else
+    return scale(r/ k , p);
+}
+
+
+/* output */
+
+
+
+
+static void rotframe(double azimuth, double elevation, double angle);
+static void rotarray(Vector *new, Vector *old, int n);
+static int mod (int i, int j);
+
+
+static void
+push_point (polyhedron *p, Vector v)
+{
+  p->points[p->npoints].x = v.x;
+  p->points[p->npoints].y = v.y;
+  p->points[p->npoints].z = v.z;
+  p->npoints++;
+}
+
+static void
+push_face3 (polyhedron *p, int x, int y, int z)
+{
+  p->faces[p->nfaces].npoints = 3;
+  Malloc (p->faces[p->nfaces].points, 3, int);
+  p->faces[p->nfaces].points[0] = x;
+  p->faces[p->nfaces].points[1] = y;
+  p->faces[p->nfaces].points[2] = z;
+  p->nfaces++;
+}
+
+static void
+push_face4 (polyhedron *p, int x, int y, int z, int w)
+{
+  p->faces[p->nfaces].npoints = 4;
+  Malloc (p->faces[p->nfaces].points, 4, int);
+  p->faces[p->nfaces].points[0] = x;
+  p->faces[p->nfaces].points[1] = y;
+  p->faces[p->nfaces].points[2] = z;
+  p->faces[p->nfaces].points[3] = w;
+  p->nfaces++;
+}
+
+
+
+
+static polyhedron *
+construct_polyhedron (Polyhedron *P, Vector *v, int V, Vector *f, int F,
+                      const char *name, const char *dual,
+                      const char *class, const char *star,
+                      double azimuth, double elevation, double freeze)
+{
+  int i, j, k=0, l, ll, ii, *hit=0, facelets;
+
+  polyhedron *result;
+  Vector *temp;
+
+  Malloc (result, 1, polyhedron);
+  memset (result, 0, sizeof(*result));
+
+  /*
+   * Rotate polyhedron
+   */
+  rotframe(azimuth, elevation, freeze);
+  Malloc(temp, V, Vector);
+  rotarray(temp, v, V);
+  v = temp;
+  Malloc(temp, F, Vector);
+  rotarray(temp, f, F);
+  f = temp;
+
+  result->number = P->index + 1;
+  result->name = strdup (name);
+  result->dual = strdup (dual);
+  result->wythoff = strdup (P->polyform);
+  result->config = strdup (P->config);
+  result->group = strdup (P->group);
+  result->class = strdup (class);
+
+  /*
+   * Vertex list
+   */
+  Malloc (result->points, V + F * 13, point);
+  result->npoints = 0;
+
+  result->nedges = P->E;
+  result->logical_faces = F;
+  result->logical_vertices = V;
+  result->density = P->D;
+  result->chi = P->chi;
+
+  for (i = 0; i < V; i++)
+    push_point (result, v[i]);
+
+  /*
+   * Auxiliary vertices (needed because current VRML browsers cannot handle
+   * non-simple polygons, i.e., ploygons with self intersections): Each
+   * non-simple face is assigned an auxiliary vertex. By connecting it to the
+   * rest of the vertices the face is triangulated. The circum-center is used
+   * for the regular star faces of uniform polyhedra. The in-center is used for
+   * the pentagram (#79) and hexagram (#77) of the high-density snub duals, and
+   * for the pentagrams (#40, #58) and hexagram (#52) of the stellated duals
+   * with configuration (....)/2. Finally, the self-intersection of the crossed
+   * parallelogram is used for duals with form p q r| with an even denominator.
+   *
+   * This method do not work for the hemi-duals, whose faces are not
+   * star-shaped and have two self-intersections each.
+   *
+   * Thus, for each face we need six auxiliary vertices: The self intersections
+   * and the terminal points of the truncations of the infinite edges. The
+   * ideal vertices are listed, but are not used by the face-list.
+   *
+   * Note that the face of the last dual (#80) is octagonal, and constists of
+   * two quadrilaterals of the infinite type.
+   */
+
+  if (*star && P->even != -1)
+    Malloc(hit, F, int);
+  for (i = 0; i < F; i++)
+    if ((!*star &&
+         (frac(P->n[P->ftype[i]]), frax.d != 1 && frax.d != frax.n - 1)) ||
+        (*star &&
+         P->K == 5 &&
+         (P->D > 30 ||
+          denominator (P->m[0]) != 1))) {
+      /* find the center of the face */
+      double h;
+      if (!*star && P->hemi && !P->ftype[i])
+        h = 0;
+      else
+        h = P->minr / dot(f[i],f[i]);
+      push_point(result, scale (h, f[i]));
+
+    } else if (*star && P->even != -1) {
+      /* find the self-intersection of a crossed parallelogram.
+       * hit is set if v0v1 intersects v2v3*/
+      Vector v0, v1, v2, v3, c0, c1, p;
+      double d0, d1;
+      v0 = v[P->incid[0][i]];
+      v1 = v[P->incid[1][i]];
+      v2 = v[P->incid[2][i]];
+      v3 = v[P->incid[3][i]];
+      d0 = sqrt(dot(diff(v0, v2), diff(v0, v2)));
+      d1 = sqrt(dot (diff(v1, v3), diff(v1, v3)));
+      c0 = scale(d1, sum(v0, v2));
+      c1 = scale(d0, sum(v1, v3));
+      p = scale(0.5 / (d0 + d1), sum(c0, c1));
+      push_point (result, p);
+      p = cross(diff(p, v2), diff(p, v3));
+      hit[i] = (dot(p, p) < 1e-6);
+    } else if (*star && P->hemi && P->index != last_uniform - 1) {
+      /* find the terminal points of the truncation and the
+       * self-intersections.
+       *  v23       v0       v21
+       *  |  \     /  \     /  |
+       *  |   v0123    v0321   |
+       *  |  /     \  /     \  |
+       *  v01       v2       v03
+       */
+      Vector v0, v1, v2, v3, v01, v03, v21, v23, v0123, v0321 ;
+      Vector u;
+      double t = 1.5;/* truncation adjustment factor */
+      j = !P->ftype[P->incid[0][i]];
+      v0 = v[P->incid[j][i]];/* real vertex */
+      v1 = v[P->incid[j+1][i]];/* ideal vertex (unit vector) */
+      v2 = v[P->incid[j+2][i]];/* real */
+      v3 = v[P->incid[(j+3)%4][i]];/* ideal */
+      /* compute intersections
+       * this uses the following linear algebra:
+       * v0123 = v0 + a v1 = v2 + b v3
+       * v0 x v3 + a (v1 x v3) = v2 x v3
+       * a (v1 x v3) = (v2 - v0) x v3
+       * a (v1 x v3) . (v1 x v3) = (v2 - v0) x v3 . (v1 x v3)
+       */
+      u = cross(v1, v3);
+      v0123 = sum(v0, scale(dot(cross(diff(v2, v0), v3), u) / dot(u,u),
+                            v1));
+      v0321 = sum(v0, scale(dot(cross(diff(v0, v2), v1), u) / dot(u,u),
+                            v3));
+      /* compute truncations */
+      v01 = sum(v0 , scale(t, diff(v0123, v0)));
+      v23 = sum(v2 , scale(t, diff(v0123, v2)));
+      v03 = sum(v0 , scale(t, diff(v0321, v0)));
+      v21 = sum(v2 , scale(t, diff(v0321, v2)));
+
+      push_point(result, v01);
+      push_point(result, v23);
+      push_point(result, v0123);
+      push_point(result, v03);
+      push_point(result, v21);
+      push_point(result, v0321);
+
+    } else if (*star && P->index == last_uniform - 1) {
+      /* find the terminal points of the truncation and the
+       * self-intersections.
+       *  v23       v0       v21
+       *  |  \     /  \     /  |
+       *  |   v0123    v0721   |
+       *  |  /     \  /     \  |
+       *  v01       v2       v07
+       *
+       *  v65       v4       v67
+       *  |  \     /  \     /  |
+       *  |   v4365    v4567   |
+       *  |  /     \  /     \  |
+       *  v43       v6       v45
+       */
+      Vector v0, v1, v2, v3, v4, v5, v6, v7, v01, v07, v21, v23;
+      Vector v43, v45, v65, v67, v0123, v0721, v4365, v4567;
+      double t = 1.5;/* truncation adjustment factor */
+      Vector u;
+      for (j = 0; j < 8; j++)
+        if (P->ftype[P->incid[j][i]] == 3)
+          break;
+      v0 = v[P->incid[j][i]];/* real {5/3} */
+      v1 = v[P->incid[(j+1)%8][i]];/* ideal */
+      v2 = v[P->incid[(j+2)%8][i]];/* real {3} */
+      v3 = v[P->incid[(j+3)%8][i]];/* ideal */
+      v4 = v[P->incid[(j+4)%8][i]];/* real {5/2} */
+      v5 = v[P->incid[(j+5)%8][i]];/* ideal */
+      v6 = v[P->incid[(j+6)%8][i]];/* real {3/2} */
+      v7 = v[P->incid[(j+7)%8][i]];/* ideal */
+      /* compute intersections */
+      u = cross(v1, v3);
+      v0123 = sum(v0, scale(dot(cross(diff(v2, v0), v3), u) / dot(u,u),
+                            v1));
+      u = cross(v7, v1);
+      v0721 = sum(v0, scale(dot(cross(diff(v2, v0), v1), u) / dot(u,u),
+                            v7));
+      u = cross(v5, v7);
+      v4567 = sum(v4, scale(dot(cross(diff(v6, v4), v7), u) / dot(u,u),
+                            v5));
+      u = cross(v3, v5);
+      v4365 = sum(v4, scale(dot(cross(diff(v6, v4), v5), u) / dot(u,u),
+                            v3));
+      /* compute truncations */
+      v01 = sum(v0 , scale(t, diff(v0123, v0)));
+      v23 = sum(v2 , scale(t, diff(v0123, v2)));
+      v07 = sum(v0 , scale(t, diff(v0721, v0)));
+      v21 = sum(v2 , scale(t, diff(v0721, v2)));
+      v45 = sum(v4 , scale(t, diff(v4567, v4)));
+      v67 = sum(v6 , scale(t, diff(v4567, v6)));
+      v43 = sum(v4 , scale(t, diff(v4365, v4)));
+      v65 = sum(v6 , scale(t, diff(v4365, v6)));
+
+      push_point(result, v01);
+      push_point(result, v23);
+      push_point(result, v0123);
+      push_point(result, v07);
+      push_point(result, v21);
+      push_point(result, v0721);
+      push_point(result, v45);
+      push_point(result, v67);
+      push_point(result, v4567);
+      push_point(result, v43);
+      push_point(result, v65);
+      push_point(result, v4365);
+    }
+
+  /*
+   * Face list:
+   * Each face is printed in a separate line, by listing the indices of its
+   * vertices. In the non-simple case, the polygon is represented by the
+   * triangulation, each triangle consists of two polyhedron vertices and one
+   * auxiliary vertex.
+   */
+  Malloc (result->faces, F * 10, face);
+  result->nfaces = 0;
+
+  ii = V;
+  facelets = 0;
+  for (i = 0; i < F; i++) {
+    if (*star) {
+      if (P->K == 5 &&
+          (P->D > 30 ||
+           denominator (P->m[0]) != 1)) {
+        for (j = 0; j < P->M - 1; j++) {
+          push_face3 (result, P->incid[j][i], P->incid[j+1][i],  ii);
+          facelets++;
+        }
+
+        push_face3 (result, P->incid[j][i], P->incid[0][i],  ii++);
+        facelets++;
+
+      } else if (P->even != -1) {
+        if (hit && hit[i]) {
+          push_face3 (result, P->incid[3][i], P->incid[0][i],  ii);
+          push_face3 (result, P->incid[1][i], P->incid[2][i],  ii);
+        } else {
+          push_face3 (result, P->incid[0][i], P->incid[1][i],  ii);
+          push_face3 (result, P->incid[2][i], P->incid[3][i],  ii);
+        }
+        ii++;
+        facelets += 2;
+
+      } else if (P->hemi && P->index != last_uniform - 1) {
+        j = !P->ftype[P->incid[0][i]];
+
+        push_face3 (result, ii, ii + 1, ii + 2);
+        push_face4 (result, P->incid[j][i], ii + 2, P->incid[j+2][i], ii + 5);
+        push_face3 (result, ii + 3, ii + 4, ii + 5);
+        ii += 6;
+        facelets += 3;
+      } else if (P->index == last_uniform - 1) {
+        for (j = 0; j < 8; j++)
+          if (P->ftype[P->incid[j][i]] == 3)
+            break;
+        push_face3 (result, ii, ii + 1, ii + 2);
+        push_face4 (result,
+                    P->incid[j][i], ii + 2, P->incid[(j+2)%8][i], ii + 5);
+        push_face3 (result, ii + 3, ii + 4, ii + 5);
+
+        push_face3 (result, ii + 6, ii + 7, ii + 8);
+        push_face4 (result,
+                    P->incid[(j+4)%8][i], ii + 8, P->incid[(j+6)%8][i],
+                    ii + 11);
+        push_face3 (result, ii + 9, ii + 10, ii + 11);
+        ii += 12;
+        facelets += 6;
+      } else {
+
+        result->faces[result->nfaces].npoints = P->M;
+        Malloc (result->faces[result->nfaces].points, P->M, int);
+        for (j = 0; j < P->M; j++)
+          result->faces[result->nfaces].points[j] = P->incid[j][i];
+        result->nfaces++;
+        facelets++;
+      }
+    } else {
+      int split = (frac(P->n[P->ftype[i]]),
+                   frax.d != 1 && frax.d != frax.n - 1);
+      for (j = 0; j < V; j++) {
+        for (k = 0; k < P->M; k++)
+          if (P->incid[k][j] == i)
+            break;
+        if (k != P->M)
+          break;
+      }
+      if (split) {
+        ll = j;
+        for (l = P->adj[k][j]; l != j; l = P->adj[k][l]) {
+          for (k = 0; k < P->M; k++)
+            if (P->incid[k][l] == i)
+              break;
+          if (P->adj[k][l] == ll)
+            k = mod(k + 1 , P->M);
+          push_face3 (result, ll, l, ii);
+          facelets++;
+          ll = l;
+        }
+        push_face3 (result, ll, j, ii++);
+        facelets++;
+
+      } else {
+
+        int *pp;
+        int pi = 0;
+        Malloc (pp, 100, int);
+
+        pp[pi++] = j;
+        ll = j;
+        for (l = P->adj[k][j]; l != j; l = P->adj[k][l]) {
+          for (k = 0; k < P->M; k++)
+            if (P->incid[k][l] == i)
+              break;
+          if (P->adj[k][l] == ll)
+            k = mod(k + 1 , P->M);
+          pp[pi++] = l;
+          ll = l;
+        }
+        result->faces[result->nfaces].npoints = pi;
+        result->faces[result->nfaces].points = pp;
+        result->nfaces++;
+        facelets++;
+      }
+    }
+  }
+
+  /*
+   * Face color indices - for polyhedra with multiple face types
+   * For non-simple faces, the index is repeated as many times as needed by the
+   * triangulation.
+   */
+  {
+    int ff = 0;
+    if (!*star && P->N != 1) {
+      for (i = 0; i < F; i++)
+        if (frac(P->n[P->ftype[i]]), frax.d == 1 || frax.d == frax.n - 1)
+          result->faces[ff++].color = P->ftype[i];
+        else
+          for (j = 0; j < frax.n; j++)
+            result->faces[ff++].color = P->ftype[i];
+    } else {
+      for (i = 0; i < facelets; i++)
+        result->faces[ff++].color = 0;
+    }
+  }
+
+  if (*star && P->even != -1)
+    free(hit);
+  free(v);
+  free(f);
+
+  return result;
+}
+
+
+
+/* External interface (jwz)
+ */
+
+void
+free_polyhedron (polyhedron *p)
+{
+  if (!p) return;
+  Free (p->wythoff);
+  Free (p->name);
+  Free (p->group);
+  Free (p->class);
+  if (p->faces)
+    {
+      int i;
+      for (i = 0; i < p->nfaces; i++)
+        Free (p->faces[i].points);
+      Free (p->faces);
+    }
+  Free (p);
+}
+
+
+int
+construct_polyhedra (polyhedron ***polyhedra_ret)
+{
+  double freeze = 0;
+  double azimuth = AZ;
+  double elevation = EL;
+  int index = 0;
+
+  int count = 0;
+  polyhedron **result;
+  Malloc (result, last_uniform * 2 + 3, polyhedron*);
+
+  while (index < last_uniform) {
+    char sym[4];
+    Polyhedron *P;
+
+    sprintf(sym, "#%d", index + 1);
+    if (!(P = kaleido(sym, 1, 0, 0, 0))) {
+      Err (strerror(errno));
+    }
+
+    result[count++] = construct_polyhedron (P, P->v, P->V, P->f, P->F,
+                                            P->name, P->dual_name,
+                                            P->class, "",
+                                            azimuth, elevation, freeze);
+
+    result[count++] = construct_polyhedron (P, P->f, P->F, P->v, P->V,
+                                            P->dual_name, P->name,
+                                            P->dual_class, "*",
+                                            azimuth, elevation, freeze);
+    polyfree(P);
+    index++;
+  }
+
+  *polyhedra_ret = result;
+  count++; /* leave room for teapot */
+  return count;
+}