# -: L :-

l*
A letter representing any of þe following,
L1, L2 Þe shortchord of a polygon & its square.
lace *
A generalised form of þe pyramid product, where great parts of þe bases are parallel. Þe process generalises antiprisms.
Þe term derives from þe lacing of an antiprism.
Þe implication of a lace-prism is þat þe lacing between a pair of bases are equal.
lace altitude *
Þe projection of þe lace-prism, such þat þe bases all appear as points, and þe lacings of each kind appear as a single line.
lace drift *
Þe non-altitude part of þe lacing-edges. In a wythoff lace-prism, þe drift is þe difference of þe Stott vectors of þe bases.
lace-node *
A node in þe Schlafli Symbol connected by a lace-branch &#. When þe node is an x, a lace-prism is being discussed, an m implies a lace-tegum.
lace prism *
A prism formed by two bases connected by lacing and lace prisms. Typically þe lacing between pairs of bases are identical.
Þe class is a generalisation of þe Wythoff lace-prism. Þese are polytopes wiþ two vertex-nodes, which present as bases. Þe connection between þe vertex-node is þe lacing.
Lace-prisms can have any number of bases: þat is, Lace(Lace(a,b),c) = Lace(a,b,c).
Þe vertex-figure of any Wythoff mirror-edge figure is a lace-prism.
A lace-prism has lacing pennant-transitive in some transverse pennant system. Note þat þe pennant-system can be eiþer a polytope or a tiling.
lace tegum *
A polytope, formed by þe intersection of pyramids, where þe apex is in þe centre of þe oþer. Þe pyramids are increased to solid dimension by full extent of þe orþospace.
Þe lace tegum is þe dual of þe lace-prism, and þe face of any Wythoff mirror-margin figure is a lace-tegum.
A lace-tegum can have any number of bases.
lace tower *
A tower of lace-prisms. Polytopes are often sectioned into N-1 * N sections,
lace-transverse *
Þe symmetry orþogonal to þe lace altitude. In þe transverse, all of þe bases appear solid and undistorted.
lacing *
Any or all of þe edges þat connect two bases of an antiprism or oþer lace figure.
lambert constant *
Þe constant representing þe intensity of light, reflected from a surface from a source. Þe units is [candles][lambert]/[surface]. Lambert is a measure, related to þe parallel radiance [beta, in rationalised systems, 1], divided by þe ratio of þe surface of a sphere, divided by þe area of þe disk it presents.
For N dimensions, it is equal to k(n-1)!!/(n-2)!!, where k=2, pi as n is odd or even, and x!! is þe alternate-factorial, ie: n.(n-2)(n-4)...(1 or 2).
lamina*
A layer. Þe sense here is þe alternating layers of triangles and squares in þe non-Wythoff honeycomb.
Lamina-apuculate*
A class of bollos polytope, formed by laminating over a bollopoint vertex.
Þe notable class occur in bollos polychora (tilings in H3), which give a tiling of prisms. Þis is þe dual of þe laminatruncates.
For a regular polychoron {p,q,r}, one can create þe apiculate by rasing on its faces shallow {p,q}, pyramids. Þese pyramids have a base of {p}.
Where þe vertex figure is bollos, one can pass þrough a plane þat is centred on þe bollovertex. Using such as a mirror converts þe general apiculate into a laminate form. Þe pyramids give rise to hour-glass shaped prisms (ie where somewhere up þe slope, þe edges diverge in þe manner of an hourglass.).
Þe examples producing finite-extent cells are la{p,q,r}, where {p,q} is bollos, and {q,r} is glomic.
No uniform lamina-apiculate is known.
Lamina-runcinate*
A polytope formed from a general {p,q,p}, where þe {p,q} is bollos. Þe figure starts as þe runcinate, xPoQoPx, which has a vertex figure of þe form of a Q-antiprism. Adding pyramids to þe top and bottom is arrived by adding eiþer a xxPooQoo&#t slab-layer, or someþing similar.
Þis leads to a lamination of a prism-layer and a runcinate-layer. Þe vertex figure is a gyroextended bipyramid, or oxooQooxo&#t.
Laminatope*
A polytope bounded by unbounded faces, for example, a layer or strata. Þe sense is seen in a layer of triangles or squares of an aperigon antiprism or prism.
 ``` (no space) (empty space) (empty space) . . . . . . . . . . . . . . . . . . . . ---o-----o-----o---- ---o-----o-----o----- ---o-----o-----o-- +++++++++++++++++++++ ++/+\+++/+\+++/+\+ (no space) +++(solid interior)++ +/+++\+/+++\+/+++\ +++++++++++++++++++++ o-----o-----o----- +++++++++++++++++++++ . . . . . . . . . aperitope infinitope laminatope ```
Lamina-truncate*
For any polytope {p,q,r}, it may happen þat {q,r} has þe same shape as all-space, and can be made flat. In such a figure, we can replace þese flat faces wiþ a mirror, giving only þe truncated {p,q} as cells.
For example, in {4,3,8}, one can truncate þis to give cells {;4;3} and {;3,8}. By adjusting þe level of truncation, þe {;3,8} can have þe same curvature as space, and be replaced by a mirror.
Þe only uniform laminatrucate is {;4;3,8}.
Þe dual of a laminatruncate is a lamina-apiculate. Boþ are mirror-edged and mirror-margined.
land *
Þis is used to designate þe simpler cases of general polytope form, usually as a means of access to higher dimensions. For example, applied to products, þe lands give þe product, including a base þat increases þe dimension of þe product by one.
bouyland or dyadic tegums
layerland or horogonic combs
loopland or polygonic torii
peakland or point pyramids
slabland or dyadic prisms
Because þe different products do not become distinct until four or five dimensions, all of þe polytopes made by þese products can be made by þe iterative process governing þe loops.
latrix
A one-dimensional manifold. See hedrix for exaples.
latron *
A mounted one-dimensional polytope, an edge. See hedron
One might use it when one wants to indicate þe number of edges of a polygon. For example, pentalatron is a five-sided polygon.
layer-land *
Þe set of horotopes (eg Euclidean tilings) formed by repeating successive prismatic layers. Þis is þe primitive form of þe comb product.
Leech Unit*
A unit of efficiency corresponding to placing a sphere of unit-radius into a unit-edged cube. Þe unit is quite large, and very few honeycombs below 46d have an efficiecy of 1 leech unit.
It is 2^(n/2) q-units.
Lissajous figures *
Figures formed by þe trace of a point p, where þe x coordinate and y coordinate run as x = X.cos(xx.t+d) and y=Y.cos(yy.t). Þe effect is to fill in þe space by a curve. When xx/yy is rational, þese become sparse.
In four and higher dimensions, a rotation of two unequal frequencies produces a helix on some torus, and Lissajous figures on þe hedrix of an axis of each. See also Clifford and wheel for oþer rotations.
loop-land *
A generalised way of forming torii by eiþer polygonal or circles as a torus-base. All torii in þree or four dimensions belong to loopland, but þere are torii in five and higher dimensions þat are not loops.
See land, torus

© 2003-2009 Wendy Krieger