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J.H. Conway defined a set of polytopal operations. One out of these is a, called ambo, derives from αμβων, which is greek and means "elevate", e.g. like a pulpit. Although that still is not too descriptive, as a polytopal operator it is one of the most clear ones. In fact it just asks to re-use the former edge centers as new vertices. The new edges then occur as corresponding secants within the former faces. In fact, wrt. that former face, the new edges then are just the respective vertex figure. And right this pattern then continues hierarchically with growing dimensions: i.e. all its facets finally then are either the former's vertex figures, or the according ambifications of the former facets in turn.
When applying that ambification onto regular polytopes, it just happens to turn out to be not different from the according rectification, i.e. one out of the Wythoffian polytopes, in fact a quasiregular one still. However, as soon as going beyond those, that rectification operation generally is not applicable any longer, because that one at first asks that all vertices lie on a (hyper)sphere (i.e. the pre-image has to be isogonal at least) and the then to be chosen truncating (hyper)planes are chosen to be orthogonal to the according vertex radius. Within rectification then the truncation depth should have to be chosen such, that the intersection of those (hyper)planes with the former edges happen to coincide when being applied from either end. However, this not only asks that all edges would have the same size, it moreover asks that all edges should be of the same type only, because else that orthogonality to the vertex ray would have the wrong angle of intersection. That is, after all, rectification is applicable only if the pre-image would be isogonal (same vertex type) as well as isotoxal (same edge type).
Ambification in contrast, due to its definition by edge centers, does not rescue to any orthogonality of those vertex rays. It not even requires equal edge lengths. As such it becomes applicable to a much greater varity of polytopes. However, it shall be noted, that the centers of those edges, emanating from a former vertex, in general do not lie within a single (hyper)plane. Wrt. convex pre-images this might be solved then by taking the convex hull of the result there too. But at least wrt. Wythoffian pre-images those new sectioning facets underneath the former vertices (vertex figures) always are flat as required.
Despite this clear distinction between rectification and ambification, many authors sadly do still use the better known first word when it comes to names for the outcome of the application of the latter operation. For instance trip clearly has 2 different, non-equivalent edge types and thus the ray orthogonality would not apply. None the less, the outcome of the ambification of trip, which surely should be called by "ambified trip", usually is known as "rectified trip" instead, with according OBSA retrip.
Throughout this website Wythoffian polytopes usually had been sized by taking the (then all equal) edge length for unity. Ambified polytopes however in general require for multiple edge sizes. Those then likewise could be sized such, using theier smalles edge size for unity, provided the pre-image would have had u-sized edges (i.e. length 2) instead. This is because then the herein being used according secants of the regular n-gon just happen to have the well-known size x(n). Esp. secants, produced by former regular triangles, thus become unit edges again. Under this general rescaling (each) it moreover happens, that the circumradius of ambo(P) can easily be calculated from that of P: in fact one then just gets
r(ambo(P)) = sqrt[(2 r(P))2 - 1]
*) Entries marked such indeed have an isogonal as well as isotoxal pre-image, where ambo() indeed could be replaced by rect().
–
In some Johnsonian instances, like diminishings or blends, however,
the full applicability of rectification does not require
a truely isotoxal pre-image, rather the edges still act within this process as if they still would be equivalent, mainly because they surely had been
without that diminishing or blend.
**) Same as before, however at least some of the facets of the pre-image no longer can be truely rectified in turn locally, i.e. wrt. to their subspaces only.
Thence, although the full pre-image polytope could well be rectified, onto its facets it thereby applies however as a mere ambification only.
-) The not otherwise being marked ones then clearly exist as mere ambifications only.
| dim | spherical space ambifications |
|---|---|
|
2
(up) |
ambo( u-n-o = n-gon ) = x(n)-n-o *), x(n)-sized edges only ambo( u3o = trig ) = x3o *), x-sized edges only ambo( u4o = square ) = q4o *), q-sized edges only ambo( u6o = hig ) = h6o *), h-sized edges only ambo( u8o = og ) = k8o *), k-sized edges only ambo( u10o = dec ) = a10o *), a-sized edges only ambo( u u = square ) = uo ou&#zq *), q-sized edges only ambo( u3u = hig ) = do3od&#zh *), h-sized edges only ambo( u4u = og ) = Ko4oK&#zk *), k-sized edges only ambo( u5u = dec ) = Ao5oA&#za *), a-sized edges only ambo( ou&#u = trig ) = .xo&#x *), x-sized edges only ambo( uu&#u = square ) = ouo&#qt *), q-sized edges only ambo( oQo&#u = square ) = .qq.&#q *), q-sized edges only ambo( uUu&#ut = hig ) = oddo&#ht *), h-sized edges only where: x = 1, u = 2, d = 3, U = 2u = 4 q = sqrt(2), h = sqrt(3), Q = 2q x(n) = 2 cos(π/n) k = x(8) = sqrt[2+sqrt(2)] a = x(10) = sqrt[(5+sqrt(5))/2] K = kk = u+q, A = aa = u+f |
|
3
(up) |
ambo( u3o u = trip ) = uo3ox ou&#zq = retrip -), edge types: x, q ambo( u4o u = cube ) = uo4oq ou&#zq = co *), q-sized edges only ambo( u5o u = pip ) = uo5of ou&#zq = repip -), edge types: q, f ambo( u3o3o = tet ) = o3x3o = oct *), x-sized edges only ambo( o3u3o = oct ) = x3o3x = co *), x-sized edges only ambo( u3u3o = tut ) = do3od3xo&#zh = retut -), edge types: x, h ambo( u3o3u = co ) = uo3xx3ou&#zq = reco *), edge types: x, q ambo( u3o4o = oct ) = o3x4o = co *), x-sized edges only ambo( o3u4o = co ) = x3o4q = reco *), edge types: x, q ambo( o3o4u = cube ) = o3q4o = co *), q-sized edges only ambo( u3u4o = toe ) = do3od4qo&#zh = retoe -), edge types: q, h ambo( o3u4u = tic ) = xo3oK4Ko&#zk = retic -), edge types: x, k ambo( u3o4u = sirco ) = uo3qx4ou&#zq = resirco -), edge types: x, q ambo( u3o5o = ike ) = o3x5o = id *), x-sized edges only ambo( o3u5o = id ) = x3o5f = rid *), edge types: x, f ambo( o3o5u = doe ) = o3f5o = id *), f-sized edges only ambo( u3u5o = ti ) = do3od5fo&#zh = reti -), edge types: f, h ambo( o3u5u = tid ) = xo3oA5Ao&#za = retid -), edge types: x, a ambo( u3o5u = srid ) = uo3fx5ou&#zq = resrid -), edge types: x, q, f ambo( ou3oo&#u = tet ) = .xo3.ox&#x = oct *), x-sized edges only ambo( uo3ou&#u = oct ) = oxx3xxo&#xt = co *), x-sized edges only ambo( uu3oo&#u = trip ) = ouo3xox&#qt = retrip -), edge types: x, q ambo( uu4oo&#u = cube ) = ouo4qoq&#qt = co *), q-sized edges only ambo( ouo4ooo&#ut = oct ) = .xox.4.oqo.&#xt = co *), x-sized edges only ambo( uu5oo&#u = pip ) = ouo5fof&#qt = repip -), edge types: q, f ambo( ouoo5oouo&#ut = ike ) = .xoxfo.5.ofxox.&#xt = id *), x-sized edges only |
|
4
(up) |
ambo( u3o o3u = triddip ) = uo3ox xo3ou&#zq = retdip **), edge types: x, q ambo( u3o o4u = tisdip ) = uo3ox qo4ou&#zq = retisdip -), edge types: x, q ambo( u4o o4u = tes ) = uo4oq qo4ou&#zq = rit *), q-sized edges only ambo( u3o3o u = tepe ) = uo3ox3oo ou&#zq = retepe -), edge types: x, q ambo( o3o4u u = tes ) = oo3oq4uo ou&#zq = rit *), q-sized edges only ambo( u3o3o3o = pen ) = o3x3o3o = rap *), x-sized edges only ambo( o3u3o3o = rap ) = x3o3x3o = srip *), x-sized edges only ambo( u3o3o3u = spid ) = uo3ox3xo3ou&#zq = respid **), edge types: x, q ambo( u3o3u3o = srip ) = uo3xx3ou3xo&#zq = resrip -), edge types: x, q ambo( o3u3u3o = deca ) = xo3od3do3ox&#zh = redeca **), edge types: x, h ambo( u3o3o *b3o = hex ) = o3x3o *b3o = ico *), x-sized edges only ambo( o3u3o *b3o = ico ) = x3o3x *b3x = rico *), x-sized edges only ambo( u3o3o4o = hex ) = o3x3o4o = ico *), x-sized edges only ambo( o3u3o4o = ico ) = x3o3x4o = rico *), x-sized edges only ambo( o3o3u4o = rit ) = o3x3o4q = rerit *), edge types: x, q ambo( o3o3o4u = tes ) = o3o3q4o = rit *), q-sized edges only ambo( u3o3o4u = sidpith ) = uo3ox3qo4ou&#zq = residpith -), edge types: x, q ambo( o3u3u4o = tah ) = xo3od3do4oq&#zh = retah -), edge types: x, q, h ambo( u3o3o5o = ex ) = o3x3o5o = rox *), x-sized edges only ambo( o3u3o5o = rox ) = x3o3x5o = srix *), x-sized edges only ambo( o3o3u5o = rahi ) = o3x3o5f = rerahi *), edge types: x, f ambo( o3o3o5u = hi ) = o3o3f5o = rahi *), f-sized edges only ambo( u3o3o5u = sidpixhi ) = uo3ox3fo5ou&#zq = residpixhi -), edge types: x, q, f ambo( o3u3u5o = xhi ) = xo3od3do5of&#zh = rexhi -), edge types: x, f, h ambo( u3o4o3o = ico ) = o3x4o3o = rico *), x-sized edges only ambo( o3u4o3o = rico ) = x3o4q3o = rerico *), edge types: x, q ambo( u3o4o3u = spic ) = uo3ox4xo3ou&#zq = respic **), edge types: x, q ambo( o3u4u3o = cont ) = xo3oK4Ko3ox&#zk = recont **), edge types: x, k ambo( s3s4o3o = sadi ) = risadi *), x-sized edges only ambo( ou3oo3oo&#u = pen ) = .xo3.ox3.oo&#x = rap *), x-sized edges only ambo( oo3ou3oo&#u = octpy ) = .ox3.xo3.ox&#x = octaco *), x-sized edges only ambo( uu3oo3oo&#u = tepe ) = ouo3xox3ooo&#qt = retepe -), edge types: x, q ambo( uo3ou3oo&#u = rap ) = oxx3xxo3oox&#xt = srip *), x-sized edges only ambo( uo3oo3ou&#u = hex ) = oxo3xox3oxo&#xt = ico *), x-sized edges only ambo( ooo3ouo3ooo&#ut = hex ) = .oxo.3.xox.3.oxo.&#xt = ico *), x-sized edges only ambo( ou3oo4oo&#u = octpy ) = .xo3.ox4.oo&#x = octaco *), x-sized edges only ambo( oo3oo4uu&#u = tes ) = ooo3qoq4ouo&#qt = rit *), q-sized edges only ambo( ouo3ooo4ooo&#ut = hex ) = .xox.3.oxo.4.ooo.&#xt = ico *), x-sized edges only ambo( uou3ouo4ooo&#ut = ico ) = oxxxo3xxoxx4ooqoo&#xt = rico *), x-sized edges only ambo( ou3oo5oo&#u = ikepy ) = .xo3.ox5.oo&#x = ikaid *), x-sized edges only ambo( ouo3ooo5ooo&#ut = ite ) = .xox.3.oxo.5.ooo.&#xt = rite *), x-sized edges only ambo( idimex ) = ridimex *), x-sized edges only |
|
5
(up) |
ambo( u3o3o3o u = penp ) = uo3ox3oo3oo ou&#zq = repenp -), edge types: x, q ambo( u3o3o o3u = tratet ) = uo3ox3oo xo3ou&#zq = retratet -), edge types: x, q ambo( u3o3o3o3o = hix ) = o3x3o3o3o = rix *), x-sized edges only ambo( o3u3o3o3o = rix ) = x3o3x3o3o = sarx *), x-sized edges only ambo( o3o3u3o3o = dot ) = o3x3o3x3o = sibrid *), x-sized edges only ambo( u3o3o3o3u = scad ) = uo3ox3oo3xo3ou&#zq = rescad **), edge types: x, q ambo( o3u3o3u3o = sibrid ) = xo3ou3xx3uo3ox&#zq = resibrid **), edge types: x, q ambo( u3o3o3o4o = tac ) = o3x3o3o4o = rat *), x-sized edges only ambo( o3u3o3o4o = rat ) = x3o3x3o4o = sart *), x-sized edges only ambo( o3o3u3o4o = nit ) = o3x3o3x4o = sibrant *), x-sized edges only ambo( o3o3o3u4o = rin ) = o3o3x3o4q = rerin *), edge types: x, q ambo( o3o3o3o4u = pent ) = o3o3o3q4o = rin *), q-sized edges only ambo( u3o3o *b3o3o = hin ) = o3x3o *b3o3o = nit *), x-sized edges only ambo( o3u3o *b3o3o = nit ) = x3o3x *b3x3o = sibrant *), x-sized edges only ambo( o3o3o *b3u3o = rat ) = o3x3o *b3o3x = sart *), x-sized edges only ambo( o3o3o *b3o3u = tac ) = o3o3o *b3x3o = rat *), x-sized edges only ambo( ou3oo3oo3oo&#u = hix ) = .xo3.ox3.oo3.oo&#x = rix *), x-sized edges only ambo( uu3oo3oo3oo&#u = penp ) = ouo3xox3ooo3ooo&#qt = repenp -), edge types: x, q ambo( uo3ou3oo3oo&#u = rix ) = oxx3xxo3oox3ooo&#xt = sarx *), x-sized edges only ambo( uo3oo3oo3ou&#u = tac ) = oxo3xoo3oox3oxo&#xt = rat *), x-sized edges only ambo( oo3oo3oo *b3ou&#u = hexpy ) = .oo3.ox3.oo *b3.xo&#x = hexaico *), x-sized edges only ambo( ou3oo3oo4oo&#u = hexpy ) = .xo3.ox3.oo4.oo&#x = hexaico *), x-sized edges only ambo( ouo3ooo3ooo4ooo&#u = tac ) = .xox.3.oxo.3.ooo.4.ooo.&#xt = rat *), x-sized edges only ambo( uo3oo4oo3ou&#u = icoap ) = oxo3xoo4oox3oxo&#xt -), x-sized edges only |
|
6
(up) |
ambo( u3o3o3o3o3o = hop ) = o3x3o3o3o3o = ril *), x-sized edges only ambo( o3u3o3o3o3o = ril ) = x3o3x3o3o3o = sril *), x-sized edges only ambo( o3o3u3o3o3o = bril ) = o3x3o3x3o3o = sabril *), x-sized edges only ambo( u3o3o3o3o3u = staf ) = uo3ox3oo3oo3xo3ou&#zq = restaf **), edge types: x, q ambo( u3o3o *b3o3o3o = hax ) = o3x3o *b3o3o3o = brox *), x-sized edges only ambo( o3u3o *b3o3o3o = brox ) = x3o3x *b3x3o3o = saborx *), x-sized edges only ambo( o3o3o *b3u3o3o = brag ) = o3x3o *b3o3x3o = siborg *), x-sized edges only ambo( o3o3o *b3o3u3o = rag ) = o3o3o *b3x3o3x = srog *), x-sized edges only ambo( o3o3o *b3o3o3u = gee ) = o3o3o *b3o3x3o = rag *), x-sized edges only ambo( u3o3o3o3o4o = gee ) = o3x3o3o3o4o = rag *), x-sized edges only ambo( o3u3o3o3o4o = rag ) = x3o3x3o3o4o = srog *), x-sized edges only ambo( o3o3u3o3o4o = brag ) = o3x3o3x3o4o = siborg *), x-sized edges only ambo( o3o3o3u3o4o = brox ) = o3o3x3o3x4o = saborx *), x-sized edges only ambo( o3o3o3o3u4o = rax ) = o3o3o3x3o4q = rerax *), edge types: x, q ambo( o3o3o3o3o4u = ax ) = o3o3o3o3q4o = rax *), q-sized edges only ambo( u3o3o3o3o *c3o = jak ) = o3x3o3o3o *c3o = rojak *), x-sized edges only ambo( o3u3o3o3o *c3o = rojak ) = x3o3x3o3o *c3o = sirjak *), x-sized edges only ambo( o3o3u3o3o *c3o = ram ) = o3x3o3x3o *c3x = sram *), x-sized edges only ambo( o3o3o3o3o *c3u = mo ) = o3o3x3o3o *c3o = ram *), x-sized edges only ambo( ou3oo3oo3oo3oo&#u = hop ) = .xo3.ox3.oo3.oo3.oo&#x = ril *), x-sized edges only ambo( uo3oo3oo3oo3ou&#u = gee ) = oxo3xoo3oo3oox3oxo&#xt = rag *), x-sized edges only ambo( oo3oo3oo *b3oo3ou&#u = tacpy ) = .oo3.oo3.oo *b3.ox3.xo&#x = taccarat *), x-sized edges only ambo( ooo3ooo3ooo *b3ooo3ouo&#u = gee ) = .ooo.3.ooo.3.ooo. *b3.oxo.3.xox.&#xt = rag *), x-sized edges only ambo( ou3oo3oo3oo4oo&#u = tacpy ) = .xo3.ox3.oo3.oo4.oo&#x = taccarat *), x-sized edges only ambo( ouo3ooo3ooo3ooo4ooo&#u = gee ) = .xox.3.oxo.3.ooo.3.ooo.4.ooo.&#xt = rag *), x-sized edges only |
| dim | flat euclidean ambifications |
|
2
(up) |
ambo( u3o6o = trat ) = o3x6o = that *), x-sized edges only ambo( o3u6o = that ) = x3o6h = rethat *), edge types: x, h ambo( o3o6u = hexat ) = o3h6o = that *), h-sized edges only ambo( u4o4o = squat ) = o4q4o = squat *), q-sized edges only ambo( o4u4o = squat ) = q4o4q = squat *), q-sized edges only ambo( u4o4u = squat ) = uo4qq4ou&#zq = squat *), q-sized edges only ambo( o3u3o3*a = trat ) = x3o3x3*a = that *), x-sized edges only |
|
3
(up) |
ambo( u4o3o4o = chon ) = o4q3o4o = rich *), q-sized edges only ambo( o4u3o4o = rich ) = q4o3x4o = rerich *), edge types: x, q ambo( o4u3u4o = batch ) = qo4od3do4oq&#zh = rebatch **), edge types: q, h ambo( u3o3o *b4o = octet ) = o3x3o *b4o = rich *), x-sized edges only ambo( o3u3o *b4o = rich ) = x3o3x *b4q = rerich *), edge types: x, q ambo( o3o3o *b4u = chon ) = o3q3o *b4o = rich *), q-sized edges only ambo( o3u3o3o3*a = octet ) = x3o3x3o3*a = rich *), x-sized edges only ambo( o3u3u3o3*a = cytatoh ) = xo3od3do3ox3*a&#zh = recytatoh **), edge types: x, h ambo( u3o3u3o3*a = rich ) = uo3xx3ou3xx3*a&#zq = rerich *), edge types: x, q ambo( gytoh ) = rigytoh -), x-sized edges only |
|
4
(up) |
ambo( u4o3o3o4o = test ) = o4q3o3o4o = rittit *), q-sized edges only ambo( o4u3o3o4o = rittit ) = q4o3x3o4o = rerittit *), edge types: x, q ambo( o4o3u3o4o = icot ) = o4x3o3x4o = ricot *), x-sized edges only ambo( u3o3o *b3o4o = hext ) = o3x3o *b3o4o = icot *), x-sized edges only ambo( o3u3o *b3o4o = icot ) = x3o3x *b3x4o = ricot *), x-sized edges only ambo( o3o3o *b3u4o = rittit ) = o3x3o *b3o4q = rerittit *), edge types: x, q ambo( o3o3o *b3o4u = test ) = o3o3o *b3q4o = rittit *), q-sized edges only ambo( u3o3o *b3o *b3o = hext ) = o3x3o *b3o *b3o = icot *), x-sized edges only ambo( o3u3o *b3o *b3o = icot ) = x3o3x *b3x *b3x = ricot *), x-sized edges only ambo( u3o3o4o3o = hext ) = o3x3o4o3o = icot *), x-sized edges only ambo( o3u3o4o3o = icot ) = x3o3x4o3o = ricot *), x-sized edges only ambo( o3o3u4o3o = bricot ) = o3x3o4q3o = rebricot *), edge types: x, q ambo( o3o3o4u3o = ricot ) = o3o3q4o3x = rericot *), edge types: x, q ambo( o3o3o4o3u = icot ) = o3o3o4x3o = ricot *), x-sized edges only ambo( o3o3o4s3s = sadit ) = risadit *), x-sized edges only |
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