# -: H :-

ha*
The ratio of the short chord of a heptagon, to its side. This is the 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 the Dynkin Symbol.
The symmetry of reflective groups in H2 may assume any polygonal shape. The hatch loop represents the dual of this region, with nodes representing cell walls, and branches representing corners.
Because there are ordinary Wythoff loops, one designates the hatch loop by placing the loop node at the front of the symbol, zb z3o3o3o3o.
hb *
The ratio if the long chord of a heptagon, to its side. This is the 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 three dimensional words, such as
• angulohedron a 2d surtope incident on another surtope.
• cohedric lying in the 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 cloth 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

The sense of 3d comes from a shape covered in a 2d cloth.
• bolohedron 3d-bollosphere, covered in H2-shaped bollohedrix
• glomohedron 3d-sphere, covered in sphere-shaped glomohedrix
• hedrous Having general extent in 2d, and other 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.

The stem can be replaced by any of the following to make higher dimensional equivalents:
• sol generic - has no particular dimension
• teel 0d cloth = button, tack, etc [Guy Inchbald]
• latr 1d cloth = string, cord, wire, thread
• hedr 2d cloth = paper, sheeting, cloth
• chor 3d cloth = clay, etc [ George Olshevsky]
• ter 4d cloth [Wendy Krieger < metric]
• pet 5d cloth [Wendy Krieger < metric]
• ect 6d cloth [Wendy Krieger < metric exa -> ecta ]
• zett 7d cloth [Wendy Krieger < metric]
• yott 8 d cloth [Wendy Krieger < metric]
See Bowers and metric thousand rules for further discussion on this.
hedrage *
The extent of two-dimensional manifold, as a measure, ie 2-length.
In higher dimensions hedrage does not have the usefulness of area. It is therefore best to avoid the use of [stem area] for this measure.
See for discussion on hedrage vs surface content.
hedrid *
solid in two dimensions.
A hexagon is hedrid always, regardless of all-space.
Note that 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 the inference that there are extra dimensions present.
Solid is now read as sol + id.
hedrix *
A two-dimensional manifold or cloth. The prefix gives an indication of the shape. One might cut hedra from hedrix, and stick them together.
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 the 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. The notion here is that one cuts things from hedrix and stick them together.
The word dodecahedron means twelve seats, is taken to mean the twelve faces. It does not matter what the balance of the figure is: the rhombic and pentagonal examples show this.
Likewise, we form to higher dimensions, other 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 that can be manifolds of any order. A ray is then described in terms of the tip and cross-section. A knife in three dimensions is a lateral hedroray, since its tip is a line, and the cross section is a 2d angle.
hedrous *
Having two general, and several minor dimensions. Hedrous timber, for example would have two large dimensions, and other as thickness. One might also describe hedrous extents of water in higher dimensions.
A journey from A to B is always latrous, although it might not lie in a line. So also hedrous, etc.
heptagon*
A regular polygon with seven sides, the smallest not to be constructed by classical methods (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
The heptagon version of the pentagonal fibonacci series. This is an iterative series that spreads over a plane. In one sector, the numbers converge to the shortchord and longchords of the heptagon. Here is a sample of a small region near the origion. As in the pentagon, this 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 the 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) lograthmetic ```
In the units-flat, the power of a^n b^m can be read straight from the table since n,m points to x in x+ya+zb, so a^5.b^3 = 23+42a+52b. This mirrors the use of the phi^n = F(n-1) + F(n)phi.

When x,y,z assume the lengths of the chords, the resultant flat is perfectly logarithmic.

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

The 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. The cube of the number -1+a+b is 7ab.

hight
To have the name of, to be called. Compare German cognate heißen.
hole*
If, in a region, there exists some sphere-surface, that can not be made to vanish, either by going to zero or infinity, a hole exists. Such a hole can be spanned by a space orthogonal to the sphere-surface, to prevent any such sphere forming in that region.
hole polynomial*
A surtope-equation like form, representing the dimensional span of surface and interior holes.
homo *
Being alike. In the context of the polygloss, it is taken that the incident flags match. An isogonal polytope has the same vertex figure, but not necessarily any higher surtope. See also iso and equi-.
honeycomb*
Aperitope. The term refers to the section of a bee's honey comb, which resembles a hexagonal tiling in 2D. In practice, it's a tiling of a slab with hexagonal prisms, capped by a rhombic dodecahedral cap.
horizon*
The notional surface where points at infinity reside. The place where it is no longer significant to discuss points in a line. Also, any points far away.
• The shape of the horizon is a complete horosphere. The geometry is therefore Euclidean.
• Every sphere on the horizon-sphere is a Eulidean plane. Such planes are only visiable from the points they pass through.
• Every point on the horizon-sphere is on the horizon of every other point.
• The inversion of circles through a point give rise to normal euclidean geometry. In practice, these giant horospheres pass through the 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 the arc of a horochord. Horodistances are the base form of measurement in circle-drawing theory.
horochorix
A 3D manifold of zero curvature or Euclidean. Such is the space we live in.
Such also occur in the surface of the horoglome or horochoron.
horogon *
The polygon {w4}, such as having zero-curvature. Its normal symbol is U.
This is sometimes called an apeirogon, but the PG distinguishes between zero-curvature [horo] and tiling [apeiro], since a tiling in hyperbolic space is less than zero curvature.
horohedrix *
A 2D manifold of zero curvature. Note that these are planes in Euclidean spaces only. See hedrix.
horopoint
A point at the horizon. These manifest themselves as points where parallel lines converge, and also the centre of the horocycle.
hororay *
A line connecting a real point to a point on the horizon. Hororays that share the same point on the horizon are parallel in one of the senses of that word.
horos *
An adjective describing the polytopes that follow a horosphere. A polytope {3,6} is always horos, regardless of whether it is a euclidean tiling or a hyperbolic polyhedron.
horosphere
A sphere of zero curvature. In Euclidean geometry, this gives a flat plane.
In hyperbolic geometry, this 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.
The centre of a horosphere is a horocentre
horotope
This term was defined as a horos polytope.
It is best avoided. A horohedron is a solid bounded by a horohedrix, but the general member of the series horohedron, horochoron, etc is a horosphere.
When one wants to designate a polytope following a horosphere, the current preferred term is horos polytope.
Hososnub *
A snub formed by alternating every vertex: that is, by replacing every vertex of a figure with a face of its vertex figure.
Such always doubles the density. It may become a compound of two snubs if the vertices are alternatable, or become a single figure otherwise. For example, the hososnub of the pentagon and hexagon give the pentagram and hexagram, respectively unique and a compound.
Professor Johnson invented the term.
hosotope*
H.S.M. Coxeter's term for a polytope with two vertices. Such are the duals to ditopes.
In spheric geometry, nullitopes exist in sparse form when the two vertices are polar, or in every geometry when the edges are not straight.
The word appears to be derived from hose, which it vaguely resembles.
hotel*
The 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 the square surhedra of a bi-pentagon prism.
Presumably the word arises from a grid-like array of rooms in a hotel.
hull*
The smallest convex figure that includes all of a set of points. Note there is no requirements for all named points to fall on the surface. Hulls are often used in stellations to describe the outer limit.
hyper- *
[Hyper] means over or above.
The sense used here is that hyperspace is a dimension usually next above all-space. For example, a theorm in 2d that involes the 3d would be invoking hyperspace, since 3d is not part of 2d.
The sense of [hyper-] in hypercube for tesseract appears from higher dimensions, in much the same way that calling a square a hyperline.
The 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, both meaning the tegum and prism products of polygons.
hyperdistance *
A measure proposed by John Conway, corresponding to exp(cish(x)). This has the feature in hyperbolic space, that the continuation on a straight line is simply the product of the two distances.
hyperspace *
A space over all-space. In practice, this is taken to have the same curvature such that all-space makes a dividing plane.
The sense of 4D space over-three dimensions is to be discouraged. From 6D, this makes hyperspace appear in the same sense as we see a plane.