There are 46 isogonal polychora that have one or more of the pennic family symmetries, they are listed below with animations of their sections, verf, and net. Animations done with Stella4D.

*These have pennic symmetry variants but no decaic versions, you may be familiar with these. Above are the nets in the same order.*

**Pen** - the pentachoron has five tets as cells and a tet for a verf, it is self dual. It has five vertices, 10 edges, and 10 equits (equalateral triangles). It is both isogonal and isotopal under pennic and kipennic symmetries. There's also a gyropennic symmetry version.

**Rap** - the rectified pentachoron has five octs and five tets and its verf is a trip. It has 10 vertices, 30 edges of one type, and its faces are equits of two types (20 + 10). It is isogonal under pennic and kipennic symmetries.

**Tip** - the truncated penteract has five tuts and five tets and its verf is a triangular pyramid. It has 20 vertices, 10+30 edges, and its faces are 20 equits and 10 hexagons/ditrigons. It is isogonal under pennic and kipennic symmetries with one degree of variance.

**Srip** - the small rhombated (or cantellated) pentachoron has five octs, five co/ratets (rhombitetratetrahedron - tetrahedral co variant), and five trips for cells. Its verf is a wedge. It has 10 vertices, 30 edges of one type, and its faces are equits of two types (20 + 10). It has 30 vertices, 30+60 edges, and its faces are 10+20+20 equits and 30 squares/rectangles. It is isogonal under pennic and kipennic symmetries with one degree of variance.

**Grip** - the great rhombated penteract has five tuts, five gratets (tetrahedral variants of toe), and ten trips. Its verf is a scalene tet. It has 60 vertices, 30+30+60 edges, and its faces are 20 equits, 10+20 ditrigons, and 30 rectangles. It is isogonal under pennic and kipennic symmetries with two degrees of variance.

**Prip** - the prismatorhombated penteract has five tuts, five ratets, 10 trips, and 10 hips/ditrips (ditrigon prism). Its verf is a trapezoid pyramid. It has 60 vertices, 30+60+60 edges, and its faces are 20+20 equits, 30+30 rectangles, and 20 ditrigons. It is isogonal under pennic and kipennic symmetries with two degrees of variance.

*These polychora can have decaic symmetry and have pennic variants, these may look familiar also. Above are the nets of deca, spid, and gippid.*

**Deca** - the decachoron has ten tuts which split into 5+5 tuts in the pennic variants. Its verf is a disphenoid (tetragonal for decaic version and rhombic for the pennic versions). It has 30 vertices and 60 edges which split into 30+30 edges in the pennic variants. Faces are 20 equits and 10 hexagons, in the pennic variants they become 10+10 equits and 10 ditrigons. It is isogonal and isotopal under decaic, kidecaic, and iodecaic symmetries with no variants. It is also isogonal but not isotopal under pennic and kipennic symmetries with one degree of variance. We could call the pennic version a pentapentachoron or 'pup' for short.

**Spid** - the small prismatodecachoron has ten tets and 20 trips for cells which splits into 5+5 tets and 10+10 trips in the pennic versions. Its verf is a trap (triangle antiprism) which becomes a triangular antifustrum in the pennic variants. It has 20 vertices and 60 edges which split into 30+30 edges in the pennic versions. Its faces are 40 equits and 30 squares in the decaic version and 20+20 equits and 30 rectangles under the pennic variants. It is isogonal under decaic, kidecaic, and iodecaic symmetries with no variants. It is also isogonal under pennic and kipennic symmetry with one degree of variance.

**Gippid** - the great prismatodecachoron has ten toes/gratets and 20 hips/ditrips. The pennic versions have 5+5 gratets and 10+10 ditrips. Its verf is an irregular tet for the pennic versions and a skew disphenoid under decaic symmetry. It has 120 vertices and 240 edges which split into 120+120 edges under pennic symmetry. Its faces are 20 hexagons, 40 ditrigons, 30 squares, and 60 rectangles. Under the pennic versions, this becomes 20+20+20 ditrigons and 30+30+30 rectangles. It is isogonal under decaic and iodecaic symmetries with one degree of variance and is isogonal under pennic symmetry with three degrees of variance. Note that it is not isogonal under kipennic nor kidecaic symmetries.

**Snad** - the snub decachoron, ah a new one finally. It has 10 snits (snub tetrahedra, an ike variant), 20 traps (chiral version, called gyro traps), and 60 skewed disphenoids. The kipennic version has 5+5 snits, 10+10 gyro traps, and 60 irregular tets. It has 60 vertices and 30+60+60+120 edges. Its faces are 20+40 equits and 120+120 other trigons, these become 20+20+20 equits and 60+60+60+60 trigons under the kipennic symmetry. It is isogonal under kidecaic symmetry with one degree of variance and is isogonal under kipennic symmetry with three degrees of variance.

* These are isogonal under decaic symmetry, but not under pennic symmetry.*

**Bideca** - (bi DEK a) the bidecachoron is the dual of deca, its cells are 30 tetragonal disphenoids. It has 10 vertices and its verf is a triakis tetrahedron. It has 20 trigonic edges (edges with three faces meeting) and 20 hexagonic edges. It has 60 isot faces (isot=isosceles triangle). It is isogonal under decaic, kidecaic, and iodecaic symmetries with no variations. It is also isotopal under decaic, kidecaic, and iodecaic symmetries as well as under pennic and kipennic where it has one degree of variations. It is the convex hull of the compound of two pens.

**Apid** - (AP id) antiprismatodecachoron. Its cells are 10 tets and 20 traps. It has 20 vertices and 60 trigonic plus 30 tetragonic edges. Its faces are 40 equits and 60 isots. The dualcell (cell of the dual) has 9 faces. Its verf is a trip symmetric saucer thing. Apid is isogonal under decaic, kidecaic, and iodecaic symmetries with no variations. It is the convex hull of the compound of two raps.

**Bited** - (BI tid) bitruncatodecachoron. Its cells are 10 tets, 30 tetragonal disphenoids, 60 disphenoids, and 20 traps. It has 40 vertices with a 10 sided dualcell. It has 60+120 tetragonic and 20 hexagonic edges. Its faces are 40 equits and 120x2 isots. It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of variance. It is a convex hull of the compound of two tips.

**Biped** - (BI ped) biprismatodecachoron. Its cells are 10 tets, 20 trips, and 30 rectas (rectangle alterprisms also called rectangular trapezoprisms or rectangular antifustrums). It has 60 trigonic edges, 60 pentagonic edges, and 20 hexagonic edges. It has 40 vertices with a 7 faced dualcell. Its faces are 40 equits, 60 rectangles, and 60 trapezoids. It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of freedom. It is the convex hull of the compound of two pennic spid variants.

**Bimted** - (BIM ted) bimesotruncatodecachoron. Its cells are 10 tuts and 20 ditras (ditrigonal alterprisms). It has 60 vertices with a five faced dualcell. It has 60 trigonic and 30+60 tetragonic edges. Its faces are 20 triangles, 60 trapezoids, and 40 ditrigons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of freedom. It is the convex hull of two pups (pennic deca variants).

**Respid** - (RES pid) rectified spid. It has 10 octs, 20 traps, and 20 retrips (rectified trips). It has 60 vertices with an 8 faced dualcell. It has two sets of 120 trigonic edges. Its faces are two sets of 40 equits, 120 isots, and 30 squares. It is isogonal under decaic, kidecaic, and iodecaic symmetries with no degree of variants and is a transitional case between mabred and sabred. It is the convex hull of two srips.

**Redeca** - (re DEK ka) rectified decachoron. It has 10 retuts (rectified tuts) and 30 disphenoids. It has 60 vertices with a wedge shaped verf and a 5 faced dualcell. It has 60+120 trigonic edges and its faces are 20 equits, 120 isots, and 20 hexagons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with no variation and it is a transitional case between sabred and gabred. It is also a convex hull of two srips.

**Mabred** - (MABE rid). medial birhombatodecachoron. It has 10 octs, 20 traps, 20 trips, 30 disphenoids, and 60 rectpyrs (rectangle pyramids). It has 60 vertices with a 10 faced dualcell. It has 60+120+120 edges, all tetragonic. Its faces are 40+40 equits, 120+120 isots, and 60 rectangles. It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of variation. It is the convex hull of two srips between respid and bideca.

**Sabred** - (SABE rid). small birhombatodecachoron. It has 10 octs, 20 traps, 30 disphenoids, and 40 trigonal antifustrums. It has 60 vertices with a 10 sided dualcell. It has 60+120 trigonic edges and 120 tetragonic edges. Its faces are 20+40+40 equits and two sets of 120 isots. It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of variation. It is the convex hull of two srips between respid and redeca and is in the middle zone of the 2-srips.

**Gabred** - (GABE rid). great birhombatodecachoron. It has 10 ratets, 20 flat traps, 30 disphenoids, and 60 rectpyrs. It has 60 vertices with a ten sided dualcell. It has 120 trigonic, 120 tetragonic, and 60 pentagonic edges. Its faces are 20+40 equits, 120+120 isots, and 60 rectangles. It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of variation. It is the convex hull of two srips in between redeca and apid.

**Sabipard** - (SAB e pard). small biprismatorhombatodecachoron. It has 10 ratets, 20 trips, 20 traps, 30 rectas, and 60 wedges. It has 120 vertices with a dualcell with 7 faces. It has 120 trigonic and 60+120+120 tetragonic edges. Its faces are 40+40 equits, 120 isots, 60+60 rectangles, and 120 trapezoids. It is isogonal under decaic, kidecaic, and iodecaic symmetries with two degrees of variations. It is the convex hull of two prips and is close to bited on the phase map.

**Mabipard** - (MAB e pard). medial biprismatorhombatodecachoron. It has 10 tuts, 20 trips, 20 traps, 30 rectas, and 40 tricus. It has 120 vertices with a 7 faced dualcell. It has two sets of 120 trigonic edges, 120 tetragonic edges, and 60 hexagonic edges. Its faces are 40+40 equits, 120 isots, 60 rectangles, 120 trapezoids, and 40 ditrigons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with two degrees of variations. It is the convex hull of two prips and is closer to sabred.

**Gabipard** - (GAB e pard). great biprismatorhombatodecachoron. It has 10 tuts, 20 ditrips (ditrigon prism), 20 traps, 30 rectas, and 60 wedges. It has 120 vertices with a 7 faced dualcell. It has 120 trigonic, 120+120 tetragonic, and 60 hexagonic edges. Its faces are 40 equits, 120 isots, 60+60 rectangles, 120 trapezoids, and 40 ditrigons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with two degrees of variations. It is the convex hull of two prips and is closer to mabred.

**Tispid** - (TIS pid). truncated spid. It has 10 tuts, 20 traps, and 20 tatrips (truncated trips). It has 120 vertices with a 5 faced dualcell. It has 60 tetragonic and two sets of 120 trigonic edges. Its faces are 40 equits, 120 isots, 40 ditrigons, and 30 ditetragons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of variations and is a transitional case. It is the convex hull of two prips and is between spid and respid as well as between mabipard and gabipard.

**Tabipard** - (TAB e pard). transitional biprismatorhombatodecachoron. It has 10 'trapezoid rectitruncated tuts', 20 trips, and 30 rectas. It has 120 vertices with a 5 faced dualcell. It has 120+120 trigonic and 60 tetragonic edges. Its faces are 40 equits, 60 rectangles, 120 trapezoids, and 20 hexagons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of variations and is a transitional case. It is the convex hull of two prips and is between redeca and spid as well as between sabipard and mabipard.

**Sobcated** - (sob CAY tid). small bicantitruncatodecachoron. It has 10 tuts, 20 trips, 20 ditras, 30 disphenoids, and 60 wedges. It has 120 vertices with a 6 faced dualcell. It has two sets of 60 and two sets of 120 tetragonic edges. Its faces are 40 equits, 120 isots, 60 rectangles, 120 trapezoids, and 40 ditrigons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with two degrees of variations. It is the convex hull of two grips and is closer to bited.

**Mobcated** - (mob CAY tid). medial bicantitruncatodecachoron. It has 10 tuts, 20 ditras, 30 disphenoids, and 40 tricus. It has 120 vertices with a 6 faced dualcell. It has 60+120 trigonic, 120 tetragonic, and 60 pentagonic edges. Its faces are 40 equits, 120 isots, 120 trapezoids, and 20+40 ditrigons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with two degrees of variations. It is the convex hull of two grips and is closer to sabred.

**Gobcated** - (gob CAY tid). great bicantitruncatodecachoron. It has 10 gratets (toe variant), 20 ditras, 30 disphenoids, and 60 wedges. It has 120 vertices with a 6 faced dualcell. It has 120 trigonic, 120 tetragonic, and two sets of pentagonic edges. Its faces are 120 isots, 60 rectangles, 120 trapezoids, and 20+40 ditrigons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with two degrees of variations. It is the convex hull of two grips and is closer to gabred.

**Tadeca** - (ta DEK ka). truncated decachoron. It has 10 tatuts (truncated tut) and 30 disphenoids. It has 120 vertices with a 4 faced dualcell. It has 60+60+120 trigonic edges. Its faces are 120 isots, 20 ditrigons, and 20 dihexagons (semiuniform dodecagons). It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of variations and is a transitional case between deca and redeca as well as between mobcated and gobcated. It is the convex hull of two grips.

**Tabcated** - (tab CAY tid). transitional bicantitruncatodecachoron. It has 10 tuts, 20 ditras, and 20 'trapezoid rectitruncated trips'. It has 120 vertices with a 5 faced dualcell. It has 120+120 trigonic and 60 tetragonic edges. Its faces are 40 equits, 30 squares, 120 trapezoids, and 40 ditrigons. It is isogonal under decaic, kidecaic, and iodecaic symmetries with one degree of variations and is a transitional case between respid and deca as well as between mobcated and sobcated. It is the convex hull of two grips.

**Sabotid** - (sa BO tid). small biomnitruncatodecachoron. It has 10 gratets, 20 ditrips, 20 ditras, 30 rectas, and 60 rectangle trapezoprisms (rectas with rectangle pyramid symmetry). It has 240 vertices with a 5 faced dualcell. It has 120+120 trigonic and three sets of 120 tetragonic edges. Its faces are three sets of 60 rectangles, two sets of 120 trapezoids, and two sets of 40 ditrigons. It is isogonal under decaic symmetry only with three degrees of variations. It is the convex hull of two gippids closer to bited.

**Gabotid** - (ga BO tid). great biomnitruncatodecachoron. It has 10 gratets, 20 ditrips, 20 ditras, 30 rectas, and 40 ditrigon trapezoprisms (ditras with trigon pyramid symmetry). It has 240 vertices with a 5 faced dualcell. It has 120+120 trigonic and three sets of 120 tetragonic edges. Its faces are two sets of 60 rectangles, two sets of 120 trapezoids, and three sets of 40 ditrigons. It is isogonal under decaic symmetry only with three degrees of variations. It is the convex hull of two gippids closer to sabred.

**Tetbotid** - (tet BO tid). tetrahedral transitional biomnitruncatodecachoron. It has 10 zratets (trapezoidal rhombitetratetrahedra), 20 ditrips, and 30 rectas. It has 240 vertices with a 4 faced dualcell. It has four sets of trigonic edges. Its faces are two sets of 60 rectangles, 120 trapezoids, 40 ditrigons, and 20 dihexagons. It is isogonal under decaic symmetry only with two degrees of variations and is a transitional case. It is the convex hull of two gippids between sabotid and gabotid as well as between deca, spid, and redeca.

**Petbotid** - (pet BO tid). prismatotransitional biomnitruncatodecachoron. It has 10 gratets, 20 ditras, and 20 zartrips (trapezorhombitrigonal prisms). It has 240 vertices with a 4 faced dualcell. It has four sets of trigonic edges. Its faces are 60 rectangles, 120 trapezoids, two sets of 40 ditrigons, and 30 ditetragons (octagon variants). It is isogonal under decaic symmetry only with two degrees of variations and is a transitional case. It is the convex hull of two gippids between sabotid and gabotid as well as between deca, spid, and respid.

* These are isogonal under kidecaic symmetry only and are thus chiral. I also called these the 'bosids'. This is where things start to look a bit more normal - just kidding, these things are crazy looking.*

**Sabosid** - (sa BO sid). small biomnisnub decachoron. It has 10 snits (snub tetrahedra - ike variants), 20+20 gyro-traps, and 30+60+60+120+120 tets of variant sorts. It has 120 vertices with a 12 faced dualcell. It has 120 trigonic, 120 pentagonic, 60 octagonic, 60+60+60+120+120 tetragonic edges. Faces are 40+40 equits and eight types of 120 scalene triangles. It is isogonal under kidecaic symmetry with three degrees of variations. It is closer to bited. It was originally called 'Bosid 2'.

**Mebosid** - (me BO sid). medial biomnisnub decachoron. It has 10 snits, 20+20 gyro-traps, and 30+60+60+120+120+120 tets of variant sorts. It has 120 vertices with a 14 faced dualcell. It has three sets of 120 trigonic, three sets of 60 tetragonic, 120 pentagonic, 60 hexagonic, 60 octagonic, and 60 decagonic edges. Faces are 40+40 equits and ten types of 120 scalene triangles. It is isogonal under kidecaic symmetry with three degrees of variations. It is closer to gabred. It was originally called 'Bosid 1'.

**Apbosid** - (ap BO sid). antipodial biomnisnub decachoron. It has 10 snits, 20+20 gyro-traps, 40 gyro trigonal antipodia, 30 disphenoids, and 60+120+120 tets of variant sorts. It has 120 vertices with a 13 faced dualcell. It has 120x3 trigonic, 60x3+120 tetragonic, and 60+60 octagonic edges. Faces are three sets of 40 equits and eight types of 120 scalene triangles. It is isogonal under kidecaic symmetry with three degrees of variations. It is closer to sabred. It was originally called 'Bosid 4'.

**Gabosid** - (ga BO sid). great biomnisnub decachoron. It has 10 snits, 20+20 gyro-traps, and 30+60+60+120+120+120 tets of variant sorts. It has 120 vertices with a 14 faced dualcell. It has three sets of 120 trigonic, three sets of 60 tetragonic, 120 pentagonic, 60 hexagonic, 60 octagonic, and 60 decagonic edges. Faces are 40+40 equits and ten types of 120 scalene triangles. It is isogonal under kidecaic symmetry with three degrees of variations. It is closer to mabred. It was originally called 'Bosid 6'. It is almost identical to the mebosid type.

**Sigwebosid** - (SIG we BO sid). small gyrowedged biomnisnub decachoron. It has 10 snits, 20+20 gyro-traps, 60 gyrowedges (it is a rectangle atop gyrated dyad), and 60+120 tets. It has 120 vertices with a 11 faced dualcell. It has 60+120+120 trigonic, three sets of 60 tetragonic, 120 tetragonic, and 60 heptagonic edges. Faces are 40+40 equits, six types of 120 scalene triangles, and 30 rectangles. It is isogonal under kidecaic symmetry with two degrees of variations and is a transitional case. It is between sabosid and mebosid. It was originally called 'Bosid 3'.

**Gigwebosid** - (GIG we BO sid). great gyrowedged biomnisnub decachoron. It has 10 snits, 20 trips, 20 traps, 60 gyrowedges, and 30+60+120 tets. It has 120 vertices with a 11 faced dualcell. It has 120+120 trigonic, 60+60+120+120 tetragonic, and 60 octagonic edges. Faces are 40+40 equits, six types of 120 scalene triangles, and 60 rectangles. It is isogonal under kidecaic symmetry with two degrees of variations and is a transitional case. It is between sabosid and gabosid. It was originally called 'Bosid 5'.

**Tetbosid** - (TET bo sid). tetrahedral transitional biomnisnub decachoron. It has 10 gyrotuts (a snub like polyhedron with ditrigons), 20 gyro-traps, and 30+60+120 tets. It has 120 vertices with a 9 faced dualcell. It has 60+120+120 trigonic, 60+60 tetragonic, 60 pentagonic, and 60 hexagonic edges. Faces are 40 equits, five types of 120 scalene triangles, and 20 ditrigons. It is isogonal under kidecaic symmetry with two degrees of variations and is a transitional case. It is between mebosid and apbosid. It was originally called 'Bosid 7'.

**Petbosid** - (PET bo sid). prismatotransitional biomnisnub decachoron. It has 10 snits, 20 gyro-traps, 20 great gyrotrips (green cell), and 60+120 tets. It has 120 vertices with a 10 faced dualcell. It has 60+120+120+120 trigonic, 60 tetragonic, 60 pentagonic, and 60 hexagonic edges. Faces are 40+40 equits, five types of 120 scalene triangles, and 30 rectangles. It is isogonal under kidecaic symmetry with two degrees of variations and is a transitional case. It is between gabosid and apbosid. It was originally called 'Bosid 8'.

* These are isogonal under iodecaic symmetry only, these have rotoreflections.*

**Sosbid** - (SOS bid). small omnisnub bidecachoron. It has 10 snits, 20 traps, 20 gyro-traps, 30 tetragonal disphenoids, and 60+120+120 tets. It has 120 vertices with a 11 faced dualcell. It has 120 trigonic, 60+120+120 tetragonic, and 120+120 pentagonic edges. Faces are 40+40 equits, and 7 types of 120 scalene triangles. It is isogonal under kidecaic symmetry with three degrees of variations. It is near bited.

**Gosbid** - (GOS bid). great omnisnub bidecachoron. It has 10 snits, 20 traps, 20 gyro-traps, 40 gyrated trigonal antipodia, 30 tetragonal disphenoids, and 120+120 irregular tets. It has 120 vertices with a 12 faced dualcell. It has three types of 120 trigonic, 60 tetragonic, 120+120 pentagonic, and 60 hexagonic edges. Faces are three sets of 40 equits and 7 types of 120 scalene triangles. It is isogonal under kidecaic symmetry with three degrees of variations. It is near sabred.

**Tetosbid** - (teh TOS bid). tetrahedral transitional omnisnub bidecachoron. It has 10 gyrotuts, 20 gyro-traps, 30 tetragonal disphenoids, and 120 irregular tets. It has 120 vertices with an 8 faced dualcell. It has 120+120 trigonic and 60+60+120 tetragonic edges. Faces are 40 equits, 4 types of 120 scalene triangles, and 20 hexagons. It is isogonal under kidecaic symmetry with two degrees of variations and is a transitional case. It is between sosbid and gosbid and in the deca-spid-redeca triangle.

**Petosbid** - (peh TOS bid). prismatotransitional omnisnub bidecachoron. It has 10 snits, 20 traps, 20 snub trips, and 120 irregular tets. It has 120 vertices with a 9 faced dualcell. It has 120+120+120 trigonic and 60+120 tetragonic edges. Faces are 40+40 equits, 4 types of 120 scalene triangles, and 30 squares. It is isogonal under kidecaic symmetry with two degrees of variations and is a transitional case. It is between sosbid and gosbid and in the deca-spid-respid triangle.

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