# -: L :-

l*
A letter representing any of the following,
L1, L2 The shortchord of a polygon & its square.
lace *
A generalised form of the pyramid product, where great parts of the bases are parallel. The process generalises antiprisms.
The term derives from the lacing of an antiprism.
The implication of a lace-prism is that the lacing between a pair of bases are equal.
lace altitude *
The projection of the lace-prism, such that the bases all appear as points, and the lacings of each kind appear as a single line.
lace drift *
The non-altitude part of the lacing-edges. In a wythoff lace-prism, the drift is the difference of the Stott vectors of the bases.
lace-node *
A node in the Schlafli Symbol connected by a lace-branch &#. When the 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 the lacing between pairs of bases are identical.
The class is a generalisation of the Wythoff lace-prism. These are polytopes with two vertex-nodes, which present as bases. The connection between the vertex-node is the lacing.
Lace-prisms can have any number of bases: that is, Lace(Lace(a,b),c) = Lace(a,b,c).
The 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 that the pennant-system can be either a polytope or a tiling.
lace tegum *
A polytope, formed by the intersection of pyramids, where the apex is in the centre of the other. The pyramids are increased to solid dimension by full extent of the orthospace.
The lace tegum is the dual of the lace-prism, and the 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 *
The symmetry orthogonal to the lace altitude. In the transverse, all of the bases appear solid and undistorted.
lacing *
Any or all of the edges that connect two bases of an antiprism or other lace figure.
lambert constant *
The constant representing the intensity of light, reflected from a surface from a source. The units is [candles][lambert]/[surface]. Lambert is a measure, related to the parallel radiance [beta, in rationalised systems, 1], divided by the ratio of the surface of a sphere, divided by the area of the 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 the alternate-factorial, ie: n.(n-2)(n-4)...(1 or 2).
lamina*
A layer. The sense here is the alternating layers of triangles and squares in the non-Wythoff honeycomb.
Lamina-apuculate*
A class of bollos polytope, formed by laminating over a bollopoint vertex.
The notable class occur in bollos polychora (tilings in H3), which give a tiling of prisms. This is the dual of the laminatruncates.
For a regular polychoron {p,q,r}, one can create the apiculate by rasing on its faces shallow {p,q}, pyramids. These pyramids have a base of {p}.
Where the vertex figure is bollos, one can pass through a plane that is centred on the bollovertex. Using such as a mirror converts the general apiculate into a laminate form. The pyramids give rise to hour-glass shaped prisms (ie where somewhere up the slope, the edges diverge in the manner of an hourglass.).
The 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 the {p,q} is bollos. The figure starts as the runcinate, xPoQoPx, which has a vertex figure of the form of a Q-antiprism. Adding pyramids to the top and bottom is arrived by adding either a xxPooQoo&#t slab-layer, or something similar.
This leads to a lamination of a prism-layer and a runcinate-layer. The vertex figure is a gyroextended bipyramid, or oxooQooxo&#t.
Laminatope*
A polytope bounded by unbounded faces, for example, a layer or strata. The 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 that {q,r} has the same shape as all-space, and can be made flat. In such a figure, we can replace these flat faces with a mirror, giving only the truncated {p,q} as cells.
For example, in {4,3,8}, one can truncate this to give cells {;4;3} and {;3,8}. By adjusting the level of truncation, the {;3,8} can have the same curvature as space, and be replaced by a mirror.
The only uniform laminatrucate is {;4;3,8}.
The dual of a laminatruncate is a lamina-apiculate. Both are mirror-edged and mirror-margined.
land *
This is used to designate the simpler cases of general polytope form, usually as a means of access to higher dimensions. For example, applied to products, the lands give the product, including a base that increases the dimension of the product by one.
layerland or horogonic combs
loopland or polygonic torii
peakland or point pyramids
Because the different products do not become distinct until four or five dimensions, all of the polytopes made by these products can be made by the iterative process governing the 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 the number of edges of a polygon. For example, pentalatron is a five-sided polygon.
layer-land *
The set of horotopes (eg Euclidean tilings) formed by repeating successive prismatic layers. This is the primitive form of the comb product.
Leech Unit*
A unit of efficiency corresponding to placing a sphere of unit-radius into a unit-edged cube. The 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 the trace of a point p, where the x coordinate and y coordinate run as x = X.cos(xx.t+d) and y=Y.cos(yy.t). The effect is to fill in the space by a curve. When xx/yy is rational, these become sparse.
In four and higher dimensions, a rotation of two unequal frequencies produces a helix on some torus, and Lissajous figures on the hedrix of an axis of each. See also Clifford and wheel for other rotations.
loop-land *
A generalised way of forming torii by either polygonal or circles as a torus-base. All torii in three or four dimensions belong to loopland, but there are torii in five and higher dimensions that are not loops.
See land, torus