This paper serves as an introduction to a newly discovered type of polyhedron. It will briefly discuss the history of polyhedra discoveries and culminate with a discussion of the Kasparian solids which are members of a new class of solid dubbed by their discoverer, the "Rational" solids.
Cube  Foursided Pyramid 
Cube  Truncated Icosohedron 
Solid Name  Signature  Click Image to View in Archimedean 

Tetrahedron  {3,3,3}  
Cube  {4,4,4}  
Octahedron  {3,3,3,3}  
Dodecahedron  {5,5,5}  
Icosahedron  {3,3,3,3,3} 
Platonic Requirements
Archimedes, in the 3rd century BC, discovered 13 regular polyhedra made of different polygon types in which the requirement that only one polygon type be used is relaxed.
Solid Name  Signature  Click Image to View in Archimedean 

Truncated Tetrahedron  {3,6,6}  
Truncated Cube  {3,8,8}  
Truncated Octahedron  {4,6,6}  
Cuboctahedron  {3,4,3,4}  
Small Rhombicuboctahedron  {3,4,4,4}  
Truncated Cuboctahedron  {4,6,8}  
Snub Cube  {3,3,3,3,4}  
Truncated Dodecahedron  {3,10,10}  
Truncated Icosahedron  {5,6,6}  
Icosidodecahedron  {3,5,3,5}  
Small Rhombicosidodecahedron  {3,4,5,4}  
Truncated Rhombicosidodecahedron  {4,6,10}  
Snub Dodecahedron  {3,3,3,3,5} 
There is also a class of polyhedra known as the prisms and antiprisms of which there is an infinite variety. Technically, they conform to the definition of an Archimedean solid but they are not generally included with them. Prisms are formed of two squares and any other single polygon. Antiprisms are formed of three triangles and any other single polygon.
Solid Name  Signature  Click Images to View in Archimedean 

Prism  {4,4,*}  etc. 
Antiprism  {3,3,3,*}  etc. 
Archimedean Requirements
In 1619 AD, Johannes Kepler discovered two more solids by casting off the assumption that a regular polygon must be convex and also relaxing the requirement that sides shouldn't intersect. By joining pentagrams (the five pointed star known alternately as a star pentagon) three at a vertex, he discovered the Great Stellated Dodecahedron and by joining them five to a vertex, the Small Stellated Dodecahedron. These are known as the Kepler solids.
Regular Pentagon (5/1) Five vertices, five edges  Star Pentagon (5/2) Five vertices, five edges 
Solid Name  Signature  Click Image to View in Archimedean  Building one Vertex  

Great Stellated Dodecahedron  {5/2,5/2,5/2} 


Small Stellated Dodecahedron  {5/2,5/2,5/2,5/2,5/2} 

Kepler Requirements
In 1809, by allowing regular polygons to intersect each other, Louis Poinsot discovered the Great Dodecahedron which is made of 5 pentagons at each vertex, circumnavigating the vertex twice before joining up, and the Great Icosahedron which is made of 5 triangles at each vertex also circumnavigating the vertex twice before joining up. These are known as the Poinsot solids. The Poinsot solids are very closely related to the Kepler solids and they are often referred to collectively as the KeplerPoinsot solids.
Solid Name  Signature  Click Image to View in Archimedean  Building one Vertex  

Great Dodecahedron  {5,5,5,5,5}/2 


Great Icosahedron  {3,3,3,3,3}/2 

Poinsot Requirements
5 convex joins  4 convex joins 1 concave join 
From the year 1878  1947, by relaxing every requirement except that all corners be the same, 53 more solids were discovered (and rediscovered) by various mathematicians including Edmund Hess (1878), Albert Badoureau (1881), Pitsch (1881), H.S.M. Coxeter and J. C. P. Miller (1930–1932), M.S. LonguetHiggins and H.C. LonguetHiggins. These are collectively known as the nonregular star polyhedra.
Solid Name  Signature  Click Image to View in Archimedean 

Great Cubicuboctahedron  {8/3,3,8/3,4}  
Cubitruncated Cuboctahedron  {8/3,6,8}  
Stellated Truncated Hexahedron  {8/3,8/3,3}  
Great Truncated Cuboctahedron  {8/3,4,6}  
Small Ditrigonal Icosidodecahedron  {5/2,3,5/2,3,5/2,3}  
Small Icosicosidodecahedron  {6,5/2,6,3}  
Small Snub Icosicosidodecahedron  {3,5/2,3,3,3,3}  
Dodecadodecahedron  {5/2,5,5/2,5}  
Truncated Great Dodecahedron  {10,10,5/2}  
Rhombidodecadodecahedron  {4,5/2,4,5}  
Snub Dodecadodecahedron  {3,3,5/2,3,5}  
Great Ditrigonal Dodecicosidodecahedron  {10/3,3,10/3,5}  
Icositruncated Dodecadodecahedron  {10/3,6,10}  
Great Icosidodecahedron  {5/2,3,5/2,3}  
Great Truncated Icosahedron  {6,6,5/2}  
Great Snub Icosidodecahedron  {3,3,5/2,3,3}  
Small Stellated Truncated Dodecahedron  {10/3,10/3,5}  
Truncated Dodecadodecahedron  {10/3,4,10}  
Great Dodecicosidodecahedron  {10/3,5/2,10/3,3}  
Great Stellated Truncated Dodecahedron  {10/3,10/3,3}  
Great Truncated Icosidodecahedron  {10/3,4,6}  
Great Retrosnub Icosidodecahedron  {3,3,3,3,5/2}/2  
Great Ditrigonal Icosidodecahedron  {3,5,3,5,3,5}/2  
Octahemioctahedron  {6,3/2,6,3}  
Tetrahemihexahedron  {4,3/2,4,3}  
Small Cubicuboctahedron  {8,3/2,8,4}  
Cubohemioctahedron  {6,4/3,6,4}  
Great Rhombicuboctahedron  {4,3/2,4,4}  
Small Rhombihexahedron  {8,4,8/7,4/3}  
Small Dodecicosidodecahedron  {10,3/2,10,5}  
Small Rhombidodecahedron  {10,4,10/9,4/3}  
Great Icosicosidodecahedron  {6,3/2,6,5}  
Small Icosihemidodecahedron  {10,3/2,10,3}  
Small Dodecicosahedron  {10,6,10/9,6/5}  
Small Dodecahemidodecahedron  {10,5/4,10,5}  
Rhombicosahedron  {6,4,6/5,4/3}  
Great Dodecahemicosahedron  {6,5/4,6,5}  
Great Rhombihexahedron  {4,8/3,4/3,8/5}  
Ditrigonal Dodecadodecahedron  {5/3,5,5/3,5,5/3,5}  
Small Ditrigonal Dodecicosidodecahedron  {10,5/3,10,3}  
Icosidodecadodecahedron  {6,5/3,6,5}  
Snub Icosidodecadodecahedron  {3,5/3,3,3,3,5}  
Inverted Snub Dodecadodecahedron  {3,5/3,3,3,5}  
Small Dodecahemicosahedron  {6,5/3,6,5/2}  
Great Dodecicosahedron  {6,10/3,6/5,10/7}  
Great Snub Dodecicosidodecahedron  {3,5/3,3,5/2,3,3}  
Great Rhombicosidodecahedron  {4,5/3,4,3}  
Great Inverted Snub Icosidodecahedron  {3,5/3,3,3,3}  
Great Dodecahemidodecahedron  {10/3,5/3,10/3,5/2}  
Great Icosihemidodecahedron  {10/3,3/2,10/3,3}  
Small Retrosnub Icosicosidodecahedron  {3,3,3,3,3,5/3}/2  
Great Rhombidodecahedron  {4,10/3,4/3,10/7}  
Great Dirhombicosidodecahedron  {4,5/3,4,3,4,5/2,4,3/2}/2 
In 1966, Norman Johnson discovered 92 more polyhedra by relaxing the requirement that the vertices be the same. Here is a sampling of the many Johnson solids.
Solid Name  Johnson Code  

Elongated Square Cupola  J19  
Metabidiminished Icosahedron  J62  
Tridiminished Rhombicosidodecahedron  J83 
Johnson Requirements
Kasparian Solids were discovered by Raffi Jacques Kasparian in 1998 while writing a virtual polyhedra construction tool, named "Archimedean",
designed to build polyhedra where all corners match an arbitrary given definition.
Its method was to build one corner according to the given definition and then to continue to propogate corners until all unconnected edges joined with
preexisting edges.
As more and more corners were built in this manner, the prospective solid would gradually become a quasispherical shell.
If there was no way to complete the solid before sides began to intersect each other, construction would be abandoned.
On the other hand, if all edges eventually joined before any sides intersected, then the solid would be complete and would have been empirically proven to exist. During the process
of development of Archimedean, it was discovered that some unrecognized solids could be constructed if sides were allowed to traverse the quasispherical shell more than once
in their attempt to complete the solid. Kasparian solids
are the result of relaxing the tacit assumption that the sides of a polyhedron may traverse the quasispherical shell only once in an attempt to join all edges.
Sides intersect in the Kasparian solids as they do in other solids but they differ in the manner in which they intersect.
In Poinsot solids and some of the star polyhedra, sides intersect as they form a corner by circumnavigating the vertex twice before joining each other.
In many of the star polyhedra, sides intersect because the sides themselves are made from polygons with intersecting line segments.
In Kasparian solids, sides intersect each other because they circumnavigate the surface of the solid more than once before finally joining each other.
While it is easy to quantify the number of times sides circumnavigate a vertex,
it is less obvious how to quantify the number of times the sides circumnavigate a surface.
This new class of solid is dubbed "Rational" because it seems selfevident that whatever the eventual method for counting circumnavigations,
the sides certainly circumnavigate the surface a rational number of times.
Kasparian Code  Signature  Click Image to View in Archimedean (Disabled for now)  One Corner Click to Play Construction Animation  Edges  Winding Value 

K1  {3,3,4}+  12:(3, 3, 4)  3/2  
K2  {3,4,5}+  60:(3, 4, 5)  3/2  
K3  {3,4,6}+  24:(3, 4, 6)  3/2  
K4  {3,5,5}+  30:(3, 3, 5, 5)  2  
K5  {3,5,10}+  60:(3, 5, 10)  3/2  
K6  {3,6,10}+  60:(3, 6, 10) 60:(6, 10)  5/4  
K7  {4,5,6}+  60:(4, 5, 6) 60:(4, 6)  5/4  
K8  {4,5,10}+  60:(4, 5, 10) 60:(4, 10)  5/4  
K9  {3,4,8}+  24:(3, 4, 8) 24:(4, 8)  5/4  
K10  {3,4,10}+  60:(3, 4, 10) 60:(4, 10)  5/4 
Kasparian Requirements
K1  Every edge is shared by two triangles and one square. (3, 3, 4) 

K2  Every edge is shared by one triangle, one square and one pentagon. (3, 4, 5) 
K3  Every edge is shared by one triangle, one square and one hexagon. (3, 4, 6) 
K4  Every edge is shared by two triangles and two pentagons. (3, 3, 5, 5) 
K5  Every edge is shared by one triangle, one pentagon and one decagon. (3, 5, 10) 
K6  Some edges are shared by one triangle, one hexagon and one decagon. (3, 6, 10)
Other edges are shared by only one hexagon and one decagon. (6, 10) 
K7  Some edges are shared by one square, one pentagon and one hexagon. (4, 5, 6)
Other edges are shared by only one square and one hexagon. (4, 6) 
K8  Some edges are shared by one square, one pentagon and one decagon. (4, 5, 10)
Other edges are shared by only one square and one decagon. (4, 10) 
K9  Some edges are shared by one triangle, one square and one octagon. (3, 4, 8)
Other edges are shared by only one square and one octagon. (4, 8) 
K10  Some edges are shared by one triangle, one square and one decagon. (3, 4, 10)
Other edges are shared by only one square and one decagon. (4, 10) 
K1 is much like the Octahedron with 3 additional squares spanning the square spaces that naturally occur inside an Octahedron,  
=  +  
K1  Octahedron  3 squares  
and K5 is much like the Icosidodecahedron with the addition of 6 decagons spanning the decagonal spaces that naturally occur inside an Icosidodecahedron. 

=  +  
K5  Icosidodecahedron  6 decagons 
K4 can be decomposed into two known solids, the Icosahedron and the Great Dodecahedron.  
=  +  
K4  Icosahedron  Great Dodecahedron 
K3 shares the same vertices as the Cuboctahedron but connects them differently,  
≈  
K3  Cuboctahedron  
and K8 shares the same vertices as the Small Rhombicosidodecahedron but connects them differently. 

≈  
K8  Small Rhombicosidodecahedron 
K2 contains within it 5 cubes plus an inner framework of pentagons and a shell with pentagonal holes that is made of intesecting triangles,  
=  +  +  
K2  5 cubes  12 pentagons  30 intesecting triangles  
and it shares the same vertices as the Great Ditrigonal Icosidodecahedron but connects them differently. 

≈  
K2  Great Ditrigonal Icosidodecahedron 
As far as my analysis has taken me, K6, K7, K9 and K10 do not contain within themselves any convex polyhedra. They do share the same vertices as some of the nonRational polyhedra.  
≈  ,  ,  
K6  Small Icosicosidodecahedron  Small Dodecicosahedron  Small Ditrigonal Dodecicosidodecahedron  
≈  ,  ,  ,  
K7  Small Snub Icosicosidodecahedron  Rhombidodecadodecahedron  Rhombicosahedron  Icosidodecadodecahedron  
≈  
K9  Small Rhombicuboctahedron  
≈  
K10  Small Rhombicosidodecahedron 
Signature  Click Image to View in Archimedean (Disabled for now)  Analysis  Edges  Winding Value 

{3,3,5}+  = Great Dodecahedron + Icosahedron ≈ K4  30:(3, 3, 5, 5)  2  
{3,4,10/3}+  ≈ Great Rhombidodecahedron  60:(3, 4, 10/3) 60:(4, 10/3)  5/4  
{5/2,10/3,4}+  ≈ Great Rhombidodecahedron  120:(4, 5/2, 10/3)  3/2  
{3,5/2,10/3}+  ≈ Great Icosihemidodecahedron  60:(3, 5/2, 10/3)  3/2  
{3,8/3,4}+  ≈ Great Cubicuboctahedron  24:(3, 4, 8/3) 24:(4, 8/3)  5/4  
{3,10/3,6}+  ≈ Great Ditrigonal Dodecicosidodecahedron  60:(3, 6, 10/3) 60:(6, 10/3)  5/4  
{5,6,10/3}+  ≈ Great Ditrigonal Dodecicosidodecahedron  60:(5, 6, 10/3) 60:(6, 10/3)  5/4  
{4,5/2,6}+  ≈ Rhombidodecadodecahedron  60:(4, 6, 5/2) 60:(4, 6)  5/4  
{5,6,5/2}+  = Dodecadodecahedron + 10 hexagons  60:(5, 6, 5/2)  3/2  
{6,5/2,10}+  ≈ Small Icosicosidodecahedron, Small Dodecicosahedron, Small Ditrigonal Dodecicosidodecahedron, K6  60:(6, 10, 5/2) 60:(6, 10)  5/4  
{4,5/2,5}+  ≈ Ditrigonal Dodecadodecahedron contains 5 cubes  60:(4, 4, 5, 5/2)  2 
The following solids are grouped into families of identical polyhedra. They illustrate that multiple signatures can sometimes arrive at the same end result.
Family  Signature  Click Image to View in Archimedean (Disabled for now)  Analysis  Edges  Winding Value 

A  {3,3,5/2}+  = Small Stellated Dodecahedron + Great Icosahedron  30:(3, 3, 5, 5)  2  
{5/2,5/2,3}+  = Small Stellated Dodecahedron + Great Icosahedron  30:(3, 3, 5, 5)  2  
{3,5/2,5/2,5/2}+  = Small Stellated Dodecahedron + Great Icosahedron Irregularities in building point to a possible bug in Archimedean.  (3, 3, 5, 5) (3, 5, 5)  ?  
B  {3,5,3,5/2}+  ≈ Ditrigonal Dodecadodecahedron  60:(3, 5, 5/2)  3/2  
{5/2,3,5/2,5}+  ≈ Ditrigonal Dodecadodecahedron  60:(3, 5, 5/2)  3/2  
C  {3,4,4,5/2}+  ≈ Ditrigonal Dodecadodecahedron contains 5 cubes  60:(3, 4, 4, 5/2)  2  
{3,4,5/2,4}+  ≈ Ditrigonal Dodecadodecahedron contains 5 cubes  60:(3, 4, 4, 5/2)  2  
{3,5/2,4}+  ≈ Ditrigonal Dodecadodecahedron contains 5 cubes  60:(3, 4, 4, 5/2)  2  
{3,4,5/2,3,5/2}+  ≈ Ditrigonal Dodecadodecahedron contains 5 partial cubes Irregularities in building point to a possible bug in Archimedean.  (3, 4, 4, 5/2) (3, 4, 5/2)  ? 