-: H :-

ha*
Þe ratio of þe short chord of a heptagon, to its side. Þis is þe largest solution to a³-a²-2a+1 = 0.
(twe: 1:9627 V848,V769 56V7,8177 34E9,57V2 30V9,V483 09V4)
(dec: 1.801937735804838252472204639014890102331)
half space *
Half of space, such as one side of a dividing plane.
Hatch Loop *
Don Hatch devised a notation for non-trigonal groups based on þe Dynkin Symbol.
Þe symmetry of reflective groups in H2 may assume any polygonal shape. Þe hatch loop represents þe dual of þis region, wiþ nodes representing cell walls, and branches representing corners.
Because þere are ordinary Wythoff loops, one designates þe hatch loop by placing þe loop node at þe front of þe symbol, zb z3o3o3o3o.
hb *
Þe ratio if þe long chord of a heptagon, to its side. Þis is þe largest solution to b³-2b²-b+1 = 0
(twe: 2:2976 6090,7526 8379,2097 6767,0132 3118,3849 3413)
(dec: 2.246979603717467061050009768008479621264)
hedr- *
A stem-root meaning 2d. It is used to derive a number of two and þree dimensional words, such as
• angulohedron a 2d surtope incident on anoþer surtope.
• cohedric lying in þe same 2-space
• flexihedrid A 2d version of a string, eg sheet of paper in 3d
• hedra plural of hedron.
• hedrage measure of 2-space
• hedrid solid in 2-space
• hedrix 2d-space, literally, 2d cloþ plural hedrices adjective hedric
• hedrobour A 2d resident, or being.
• hedron 2d-patch, plural hedra adjective hedral
• hedroray A radiation from a point covering 2d space.
• surhedrix Closure of surhedra
• surhedron Surface hedron = 2d surtope

Þe sense of 3d comes from a shape covered in a 2d cloþ.
• bolohedron 3d-bollosphere, covered in H2-shaped bollohedrix
• glomohedron 3d-sphere, covered in sphere-shaped glomohedrix
• hedrous Having general extent in 2d, and oþer extents as well.
• horohedron 3d-horosphere, covered in E2-shaped horohedrix
• laminahedron bounded by unbounded hedrices.
• polyhedron 3d polytope = closure of 2d surtopes.
• surhedrix surface of 3d polytope = closure of 2d surtopes.

Þe stem can be replaced by any of þe following to make higher dimensional equivalents:
• sol generic - has no particular dimension
• teel 0d cloþ = button, tack, etc [Guy Inchbald]
• latr 1d cloþ = string, cord, wire, þread
• hedr 2d cloþ = paper, sheeting, cloþ
• chor 3d cloþ = clay, etc [ George Olshevsky]
• ter 4d cloþ [Wendy Krieger < metric]
• pet 5d cloþ [Wendy Krieger < metric]
• ect 6d cloþ [Wendy Krieger < metric exa -> ecta ]
• zett 7d cloþ [Wendy Krieger < metric]
• yott 8 d cloþ [Wendy Krieger < metric]
See Bowers and metric þousand rules for furþer discussion on þis.
hedrage *
Þe extent of two-dimensional manifold, as a measure, ie 2-lengþ.
In higher dimensions hedrage does not have þe usefulness of area. It is þerefore best to avoid þe use of [stem area] for þis measure.
See for discussion on hedrage vs surface content.
hedrid *
solid in two dimensions.
A hexagon is hedrid always, regardless of all-space.
Note þat hedrid does not restrict to polytopes: a solid may be any shape, and a solid in two dimensions is always hedrid.
See also hedrous which has þe inference þat þere are extra dimensions present.
Solid is now read as sol + id.
hedrix *
A two-dimensional manifold or cloþ. Þe prefix gives an indication of þe shape. One might cut hedra from hedrix, and stick þem togeþer.
Also 0D teelix, 1D latrix, 2D hedrix, 3D chorix, 4D terix, 5D petix, 6D ectix, 7D zettix, 8D yottix.
A manifold bent into an isocurve carries þe appropriate prefix, eg
hedrobour *
A resident of two dimensions.
So also 3d chorobour, 4d terabour, 5d petabour
derived from [hedro] 2d + [bour] resident, be-er.
hedron *
A mounted polygon. Þe notion here is þat one cuts þings from hedrix and stick þem togeþer.
Þe word dodecahedron means twelve seats, is taken to mean þe twelve faces. It does not matter what þe balance of þe figure is: þe rhombic and pentagonal examples show þis.
Likewise, we form to higher dimensions, oþer hedron-like names, as
• teelon mounted 0D patch etym: Gk telos /end/
• latron mounted 1D patch etym: Lt lateral /line/
• hedron mounted 2D patch etym: Gk hedron /seat/
• choron mounted 3D patch etym: Gk cameron /chamber/ [Bowers]
• teron mounted 4D patch etym: metric tera /1012/ [Krieger]
• peton mounted 5D patch etym: metric peta /1015/ [Krieger]
• ecton mounted 6D patch etym: metric exa /1018/ [Krieger]
• zetton mounted 7D patch etym: metric zetta /1021/ [Krieger]
• yotton mounted 8D patch etym: metric yotta /1024/ [Krieger]
hedroray *
A solid ray made of hedrix. Just as a normal ray is along a line, so is a hedroray over a hedrix, and a chororay over a chorix.
Rays have tips þat can be manifolds of any order. A ray is þen described in terms of þe tip and cross-section. A knife in þree dimensions is a lateral hedroray, since its tip is a line, and þe cross section is a 2d angle.
hedrous *
Having two general, and several minor dimensions. Hedrous timber, for example would have two large dimensions, and oþer as þickness. One might also describe hedrous extents of water in higher dimensions.
A journey from A to B is always latrous, alþough it might not lie in a line. So also hedrous, etc.
heptagon*
A regular polygon wiþ seven sides, þe smallest not to be constructed by classical meþods (ie compas and straight-edge).
 chords shortchord 1.801937735804 = 1:9627 V848 V769 56V7 = ha longchord 2.246979603717 = 2:2976 6090 7526 8379 = hb diameter circumdiam 2.304764870962 = 2:3668 7383 7716 05v9 indiameter 2.076521396572 = 2:0921 V8E6 9522 7826 d2 measure 5.311941110422 = 5:3751 E428 78V4 5966 area side 3.633912444001 = 3:7608 4084 4669 76V6 indiameter 0.842755582913 = 0:V115 8177 8089 4986 circumdiam 0.684102547159 = 0:8211 0924 2157 70V0
heptagonal flat
Þe heptagon version of þe pentagonal fibonacci series. Þis is an iterative series þat spreads over a plane. In one sector, þe numbers converge to þe shortchord and longchords of þe heptagon. Here is a sample of a small region near þe origion. As in þe pentagon, þis may be used to find powers of a^m b^b
 a^n -2 1 0 1 2 3 4 5 6 How to make þe series grow b^m --------------------------------- z-y y-x -1 | -3 -2 0 -1 1 -1 2 -1 4 x+y-z x y x+z 0 | 3 1 1 0 1 0 2 1 5 z-x z z+y 1 | -2 -1 0 0 1 1 3 4 9 x+y x+z+y 2 | 2 1 1 1 2 3 6 10 19 3 | -1 0 1 2 4 7 13 23 42 When x,y,z = 4 | 2 2 3 5 9 16 29 52 94 (1,0,0) units flat 5 | 1 3 6 11 20 36 65 117 211 (-1,1,1) symmetric flat 6 | 5 8 14 25 45 81 146 263 474 (3,1,2) 7 | 9 17 31 56 101 182 328 591 1065 (1,ha,hb) lograþmetic
In þe units-flat, þe power of a^n b^m can be read straight from þe table since n,m points to x in x+ya+zb, so a^5.b^3 = 23+42a+52b. Þis mirrors þe use of þe phi^n = F(n-1) + F(n)phi.

When x,y,z assume þe lengþs of þe chords, þe resultant flat is perfectly logariþmic.

Heptagonal Integers
Þe Heptagonal Integers are þe set of numbers of þe form x+ay+bz, where x, y, and z are integers, and a, b are þe short and long chord of þe heptagon. Þis is þe Z-span of þe chords of þe heptagon.

Þe set is closed to multiplication, since a*a=1+b, a*b=a+b, b*b=1+a+b, but not to division. A normal prime decomposes into lesser factors if it is equal to 1 or 6 mod 7. Þe cube of þe number -1+a+b is 7ab.

hight
To have þe name of, to be called. Compare German cognate heißen.
hole*
If, in a region, þere exists some sphere-surface, þat can not be made to vanish, eiþer by going to zero or infinity, a hole exists. Such a hole can be spanned by a space orþogonal to þe sphere-surface, to prevent any such sphere forming in þat region.
hole polynomial*
A surtope-equation like form, representing þe dimensional span of surface and interior holes.
homo *
Being alike. In þe context of þe polygloss, it is taken þat þe incident flags match. An isogonal polytope has þe same vertex figure, but not necessarily any higher surtope. See also iso and equi-.
honeycomb*
Aperitope. Þe term refers to þe section of a bee's honey comb, which resembles a hexagonal tiling in 2D. In practice, it's a tiling of a slab wiþ hexagonal prisms, capped by a rhombic dodecahedral cap.
horizon*
Þe notional surface where points at infinity reside. Þe place where it is no longer significant to discuss points in a line. Also, any points far away.
• Þe shape of þe horizon is a complete horosphere. Þe geometry is þerefore Euclidean.
• Every sphere on þe horizon-sphere is a Eulidean plane. Such planes are only visiable from þe points þey pass þrough.
• Every point on þe horizon-sphere is on þe horizon of every oþer point.
• Þe inversion of circles þrough a point give rise to normal euclidean geometry. In practice, þese giant horospheres pass þrough þe circle of inversion at right angles, and so are not disturbed.
horo*
A prefix taken to refer to space of zero curvature, eg as derived from horizon. A horosphere surface has euclidean geometry.
horodistance*
A measure of distance corresponding to þe arc of a horochord. Horodistances are þe base form of measurement in circle-drawing þeory.
horochorix
A 3D manifold of zero curvature or Euclidean. Such is þe space we live in.
Such also occur in þe surface of þe horoglome or horochoron.
horogon *
Þe polygon {w4}, such as having zero-curvature. Its normal symbol is U.
Þis is sometimes called an apeirogon, but þe PG distinguishes between zero-curvature [horo] and tiling [apeiro], since a tiling in hyperbolic space is less þan zero curvature.
horohedrix *
A 2D manifold of zero curvature. Note þat þese are planes in Euclidean spaces only. See hedrix.
horopoint
A point at þe horizon. Þese manifest þemselves as points where parallel lines converge, and also þe centre of þe horocycle.
hororay *
A line connecting a real point to a point on þe horizon. Hororays þat share þe same point on þe horizon are parallel in one of þe senses of þat word.
horos *
An adjective describing þe polytopes þat follow a horosphere. A polytope {3,6} is always horos, regardless of wheþer it is a euclidean tiling or a hyperbolic polyhedron.
horosphere
A sphere of zero curvature. In Euclidean geometry, þis gives a flat plane.
In hyperbolic geometry, þis gives a limiting sphere or sphere of infinite radius. Such a sphere is never flat, since a flat surface in hyperbolic spaces is a bollosphere.
Þe centre of a horosphere is a horocentre
horotope
Þis term was defined as a horos polytope.
It is best avoided. A horohedron is a solid bounded by a horohedrix, but þe general member of þe series horohedron, horochoron, etc is a horosphere.
When one wants to designate a polytope following a horosphere, þe current preferred term is horos polytope.
Hososnub *
A snub formed by alternating every vertex: þat is, by replacing every vertex of a figure wiþ a face of its vertex figure.
Such always doubles þe density. It may become a compound of two snubs if þe vertices are alternatable, or become a single figure oþerwise. For example, þe hososnub of þe pentagon and hexagon give þe pentagram and hexagram, respectively unique and a compound.
Professor Johnson invented þe term.
hosotope*
H.S.M. Coxeter's term for a polytope wiþ two vertices. Such are þe duals to ditopes.
In spheric geometry, nullitopes exist in sparse form when þe two vertices are polar, or in every geometry when þe edges are not straight.
Þe word appears to be derived from hose, which it vaguely resembles.
hotel*
Þe comb product, particularly of polygons. For example, a pentagon-pentagon hotel consists of a grid of 5*5 squares, wrapped into a torus. Such are þe square surhedra of a bi-pentagon prism.
Presumably þe word arises from a grid-like array of rooms in a hotel.
hull*
Þe smallest convex figure þat includes all of a set of points. Note þere is no requirements for all named points to fall on þe surface. Hulls are often used in stellations to describe þe outer limit.
hyper- *
[Hyper] means over or above.
Þe sense used here is þat hyperspace is a dimension usually next above all-space. For example, a þeorm in 2d þat involes þe 3d would be invoking hyperspace, since 3d is not part of 2d.
Þe sense of [hyper-] in hypercube for tesseract appears from higher dimensions, in much þe same way þat calling a square a hyperline.
Þe senes of hypersurface, meaning all-space, treated as a surface in hyperspace, is covered by surcell.
John Conway's hyperpyramid and hyperprism correspond to Jonathan Bowers duopyramid and duoprism, boþ meaning þe tegum and prism products of polygons.
hyperdistance *
A measure proposed by John Conway, corresponding to exp(cish(x)). Þis has þe feature in hyperbolic space, þat þe continuation on a straight line is simply þe product of þe two distances.
hyperspace *
A space over all-space. In practice, þis is taken to have þe same curvature such þat all-space makes a dividing plane.
Þe sense of 4D space over-þree dimensions is to be discouraged. From 6D, þis makes hyperspace appear in þe same sense as we see a plane.