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**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
- angulo
**hedr**on a 2d surtope incident on anoþer surtope. - co
**hedr**ic lying in þe same 2-space - flexi
**hedr**id A 2d version of a string, eg sheet of paper in 3d -
**hedr**a plural of**hedr**on. -
**hedr**age measure of 2-space -
**hedr**id solid in 2-space -
**hedr**ix 2d-space, literally, 2d cloþ plural**hedr**ices adjective**hedr**ic -
**hedr**obour A 2d resident, or being. -
**hedr**on 2d-patch, plural**hedr**a adjective**hedr**al -
**hedr**oray A radiation from a point covering 2d space. - sur
**hedr**ix Closure of sur**hedr**a - sur
**hedr**on Surface hedron = 2d surtope

Þe sense of 3d comes from a shape covered in a 2d cloþ.- bolo
**hedr**on 3d-bollosphere, covered in H2-shaped bollo**hedr**ix - glomo
**hedr**on 3d-sphere, covered in sphere-shaped glomo**hedr**ix -
**hedr**ous Having general extent in 2d, and oþer extents as well. - horo
**hedr**on 3d-horosphere, covered in E2-shaped horo**hedr**ix - lamina
**hedr**on bounded by unbounded**hedrices**. - poly
**hedr**on 3d polytope = closure of 2d surtopes. - sur
**hedr**ix 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]

- angulo
**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- glomohedrix Þe surface of a 3d sphere.
- horohedrix Þe 2D euclidean plane
- bollohedrix Þe hyperbolic 2-space
- planohedrix A flat plane

**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 /10*[Krieger]^{12}/ - peton mounted 5D patch
*etym: metric peta /10*[Krieger]^{15}/ - ecton mounted 6D patch
*etym: metric exa /10*[Krieger]^{18}/ - zetton mounted 7D patch
*etym: metric zetta /10*[Krieger]^{21}/ - yotton mounted 8D patch
*etym: metric yotta /10*[Krieger]^{24}/

- teelon mounted 0D patch
**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.

See also approach, verge. **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
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.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

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.

See also glomos, bollos **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.

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© 2003-2009 Wendy Krieger