ISSN 2590-9770 
The Art of Discrete and Applied Mathematics 4 (2021) #P3.07 
https://doi.org/10.26493/2590-9770.1354.b40 
(Also available at http://adam-journal.eu) 
Locally spherical hypertopes from generalised cubes* 
Antonio Montero† , Asia Ivi´cWeiss 
Departmentof Mathematics and Statistics,York University, Toronto, Ontario M3J 1P3, Canada 
Received 28 January 2020, accepted 31 July 2020, published online 23 August 2021 
Abstract 
We show that every non-degenerate regular polytope can be used to construct a thin, residually-connected, chamber-transitive incidence geometry, i.e. a regular hypertope. These hypertopes are related to the semi-regular polyotopes with a tail-triangle Coxeter diagram constructed by Monson and Schulte. We discuss several interesting examples de­rived when this construction is applied to generalised cubes. In particular, we produce an example of a rank 5 fnite locally spherical properhypertope ofhyperbolic type. No such examples were previously known. 
Keywords:Regularity, thingeometries, hypermaps, hypertopes, abstract polytopes. Math. Subj. Class.: 52B15, 51E24, 51G05 
Introduction 
Hypertopesareaspecialtypeof incidence geometriesthat generalisethe notionsof abstract polytopesandofhypermaps. The conceptwas introducedin[9]with particular emphasis on regularhypertopes (that is, the ones with highest degree of symmetry). Although in [8,10,11]a numberof interestingexamplesofregularhypertopeshavebeen constructed, within the theory of abstract regular polytopes much more work has been done. Notably, [26]and[28]deal with universal constructions of polytopes, while in[5, 23, 24]some con­
structions with prescribed combinatorial conditions are explored. In another direction, in [3,7,14,22]the questionsofexistenceof polytopes with prescribed (interesting) groups 
*Supported by NSERC. The authors wish to thank the anonymous referee for their useful comments. Their suggestions helped to improve the manuscript. 
†Corresponding author. E-mail addresses: amontero@yorku.ca (Antonio Montero), weiss@mathstat.yorku.ca (Asia Ivi´cWeiss) 
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 
are investigated. Much of theimpetus to the development of the theory of abstract poly­topes, as well as the inspiration with the choice of problems, was based on work of Branko Gr¨
unbaum[13]from the 1970s. 
Inthispaperwe generalisethehalving operationon polyhedra(see7Bin[18])toacer­
tain class of regular abstract polytopes to construct regular hypertopes. More precisely, given a regular non-degenerate n-polytope P, we construct a regular hypertope H(P) whose type-preserving automorphism group is a subgroup of Aut(P) of index at most 
2. Thehypertope H(P),aswe shall seein Section 3,is closely relatedtothe semi-regular polytopes with tail-triangle Coxeter diagram describedby Monson and Schultein[19,20]. 
Thepaperisorganisedasfollows.In Section 2wereviewthebasictheoryofhypertopes (with particular focus on regularhypertopes) and revisit the notionofa regular polytope (frst introduced in the early 1980s) within the theory of hypertopes. In Section 3 we explore the halving operation on an abstract polytope and show thatthe resulting incidence system is a regular hypertope. Finally, in Section 4 we give concrete examples arising from our construction. In particular, we focus on locally sphericalhypertopes arising from Danzer’sconstructionof generalised cubes.Asaresultwe produceanumberofnewregular hypertopes includinganexampleofa fniteregularrank5 properhypertope which, because of the sizeof its automorphism group, could not previouslybe found (see[11, Section 6]). 
2 Regular hypertopes 
In this section we review the defnition and basic properties of regular hypertopes. We introduce abstract polytopes as a special class of hypertopes. However, if the reader is interestedina classic and more detailed defnitionof abstract polytopes, we suggest[18, Section 2A]. 
The notionofa regularhypertopewas introducedin[9]asa common generalisationof an abstract regular polytopeandofaregularhypermap.In short,a regular hypertope is a thin, residually-connected, chamber-transitive geometry (the concepts are defned below). More details and an accountof general theory canbe foundin[2]. 
An incidence system isa 4-tuple . := (X, *, t, I) where X and I are sets, t : X › I is the type function and * is a binary relation in X called incidence. The elements of X and I are called the elements and the types of ., respectively. The cardinality of I is the rank of .. An element x is said to be of type i, or an i-element, whenever t(x)= i, for i . I. The relation * is refexive, symmetric and such that, for all x, y . X, if x * y and t(x)= t(y), then x = y. 
Afag F is a subset of X in which every two elements are incident. An element x is incident to a fag F , denoted by x * F , when x is incident to all elements of F . For a fag F the set t(F ) := {t(x) | x . F } is called the type of F . When t(F )= I, F is called a chamber. 
An incidence system . is a geometry(or an incidencegeometry)if every fag of. is contained in a chamber, that is, if all maximal fags of . are chambers. 
The residue of a fag F of an incidence geometry . is the incidence geometry .F := (XF , * F ,tF ,IF ) where XF := {x . X : x * F,x /. F }, IF := I \ t(F ), and where tF and * F are restrictions of t and * to XF and IF respectively. 
An incidence system . is thin when every residue of rank one of . contains exactly two elements.Ifan incidence geometryis thin, thengivena chamber C thereexistsexactly one chamber differing from C in its i-element. An incidence system . is connected if its 
A. Montero and A. Ivi´
cWeiss: Locally spherical hypertopesfromgeneralised cubes 
incidence graph is connected. Moreover, . is residually connected when . is connected and each residue of . of rank at least two is also connected. It is easy to see that this condition is equivalent to strong connectivity for polytopes (as defned in[18, pg. 23] and reviewed below) and the thinness is equivalent to the diamond condition for polytopes. A hypertope is a thin incidence geometry which is residually connected. 
An abstract polytope of rank n is usually defned as a strongly-connected partially or­dered set (P, 6) that satisfes the diamond condition and in such a way that all maximal chains of P have the same length(n +2). In the language of incidence geometries, an ab­stract polytope is an incidence system (P, *6, rk, {-1,...,n}), where *6 is the incidence relation defned by the order of P (i.e., x *6 y if and only if x 6 y or y 6 x)andrk is the rank function. We require that P has a unique (minimum) element of rank (type) -1 and a unique (maximum) element of rank n. Note that a fag (in the language of incidence systems) is what has been called a chain in the theory of abstract polytopes. Therefore, maximal chains of P are precisely the chambers of the corresponding incidence system. Thefact thatevery maximal chainof P has (n + 2) elements implies that P defnes a ge­ometry. It is well-known that for anytwo incident elements Fi 6 Fj of P, with rk(Fi)= i and rk(Fj)= j, the section Fj/Fi = {x .P : Fi 6 x 6 Fj} is a (j - i - 1)-polytope. We note that for polytopes, the residue of a chainF is a union of sections of P defned by the intervals of IF . 
Observe that the rank 2 hypertopes are precisely the abstract polygons and the rank3 hypertopes are the non-degeneratehypermaps. 
Atype-preserving automorphism of an incidence system . := (X, *, t, I) is a permu­tation . of X such that for every x . X, t(x)= t(x.) and if x, y . X, then x * y if and only if x. * y.. The set of type-preserving automorphisms of . is denoted by AutI (.). 
The group of type-preserving automorphisms of an incidence geometry . generalises the automorphism group of an abstract polytope. Somefamiliar symmetry properties of polytopes extend naturally to incidence geometries. For instance, AutI (.) actsfaithfully on the set of chambers of .. Moreover, if . isahypertope this actionis semi-regular. In fact, if. . AutI (.) fxes a chamber C, it also fxes its i-adjacent chamber Ci. Since . is residually connected, . must be the identity. 
We say that. is chamber-transitive if the action of AutI (.) on the chambers is tran­sitive, and in that case the action of . on the set of chambers is regular. For that reason . is then called a regular hypertope. As expected, this generalises the concept of a regular polytope. 
Observethat, when . isageometry,chamber-transitivityisequivalentto fag-transitivity (meaning that for each J . I, thereisaunique orbitonthefagsoftype J under the action of AutI (.);see for example Proposition 2.2 in[9]). 
Let . := (X, *, t, I) bea regularhypertope and let C be a fxed (base) chamber of .. For eachi . I there exists exactly one automorphism .i mapping C to Ci. If F . C is a fag, then the automorphism group of the residue .F is precisely stabiliser of F under the action of AutI (.).We denote this group by Stab.(F ). It is easy to see that 
Stab.(F )= h.i : i . IF i . 
If IF = {i}, that is .F is of rank |I|- 1, the thinness of . implies that 
.2 =1. (2.1) 
i 
If IF = {i, j}, then there exists pij .{2,..., .} such that 
(.i.j )pij = 1; (2.2) 
in this situation the residue of F is an abstract pij -gon. Moreover, if J and K are arbitrary subsets of I and F, G . C are fags such that IF = J and IG = K, then 
Stab.(F ) . Stab.(G) = Stab.(F . G), 
or equivalently h.j : j . Ji.h.k : k . Ki = h.i : i . J . Ki . (2.3) 
We call the condition in(2.3)theintersection property. Following[9], a C-group is a group generated by involutions {.i : i . I} that satisfes the intersection property. It follows thatthe type-preserving automorphism groupofaregularhypertopeisa C-group ([9, Theorem 4.1]). 
Every Coxeter group U is a C-group and in particular, it is the type-preserving auto-morphismofa regularhypertope[32, Section3] called the universal regular hypertope associated with the Coxeter group U. Moreover, every C-group G is a quotient of a Cox-eter group U. If H isa regularhypertope whose type-preserving automorphism groupis G, the universal cover of H is the regularhypertope associated with U. 
The Coxeter diagram of a C-group G is a graph with |I| vertices corresponding to the generators of G and with an edge {i, j} whenever the order pij of .i.j is greater than 2. The edge is endowed with the label pij when pij > 3. The automorphism group of an abstract polytopeisa string C-group, thatis,a C-grouphavinga linearCoxeter diagram.If P isa regular n-polytope, then we say that P isof (Schl¨afi) type {p1,...,pn-1} whenever the Coxeter diagram of Aut(P) is 
.0.2.n-2 .n-1 
... 
p1 pn-1 
One of the most remarkable results in the theory of abstract regular polytopes is that the string C-groups are precisely the automorphism groups of the regular polytopes. In other words, given a string C-group G, there exists a regular polytope P = P(G) such that G = Aut(P) (see[18, Section 2E]). This resultwasprovedin[25,27]for so-called regular incidence complexes, (combinatorial objects slightly more general than abstract polytopes). However, the results were essentially already known toTits who constructed coset geometries from string Coxeter groups in[30], which preceded the introduction of the intersection propertyinaworking paper from 1961 (see[32]). Nevertheless, Schulte was notawareof this. In[29]hegivesa nice historical note on thedevelopmentof the theory. 
Analogously,itis also possibleto construct, under certain conditions,aregularhyper­tope from a group, and particularly from a C-group, using the following proposition. 
Proposition 2.1 (Tits Algorithm[32]). Let n bea positive integer and I := {0,...,n-1}. Let G be a group together with a family of subgroups (Gi)i.I , X the set consisting of all cosets Gig with g . G and i . I, and t : X › I defned by t(Gig)= i. Defne an incidence relation * on X × X by: 
Gig1 * Gjg2 if and only if 6
Gig1 . Gj g2 = Ø. 
A. Montero and A. Ivi´
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Then the 4-tuple . := (X, *, t, I) is an incidence system having {Gi : i . I} as a cham­ber. Moreover, the group G acts by right multiplication as an automorphism group on .. Finally, the groupG is transitive on the fags of rank less than 3. 
The incidence system constructed using the proposition above will be denoted by .(G;(Gi)i.I ) and called a coset incidence system. 
Theorem 2.2 ([9, Theorem 4.6]). Let I = {0,...,n - 1}, let G = h.i | i . Ii be a C-group, and let . := .(G;(Gi)i.I ) where Gi := h.j | j = 6ii for all i . I. If G is fag-transitive on ., then . is a regular hypertope. 
In other words, the coset incidence system . = .(G, (Gi)i.I ) isa regularhypertopeif and only if the group G is a C-group and . is fag-transitive. In order to prove that a given group G is a C-group, we can use the following result. 
Proposition 2.3 ([7, Proposition 6.1]). Let G be a group generated by n involutions .0,...,.n-1. Suppose that Gi is a C-group for every i .{0,...,n - 1}. Then G is a C-group if and only if Gi . Gj = Gi,j for all 0 6 i, j 6 n - 1. 
At the end of this section we introduce Lemma 2.4 whose proof is straightforward and willbe usedin Section 3toprove our main results. 
Lemma 2.4. Let G = h.0,...,.r-1i and H = h.r,...,.r+s-1i be two C-groups. Then the group 
G × H = h.0,...,.r-1,.r,...,.r+s-1i 
is a C-group. 
Halving operation 
In this section the halving operation is applied to the automorphism group of a non-degenerate regular polytope P producing H(P), which is a subgroup of Aut(P) of index at most 2.We prove that the group H(P) isa C-group and that the corresponding incidence system is fag-transitive. Therefore the group H(P) is the type-preserving automorphism groupofa regularhypertope. 
Let n > 3 and P bea regular,non-degenerate n-polytope of type {p1,...,pn-2,pn-1} and automorphism group Aut(P)= h%0,...,%n-1i. The halving operation is the map 
. : h%0,...,%n-1i›h.0,...,.n-1i , 
where (
%i, if 0 6 i 6 n - 2, 
.i = (3.1) 
%n-1%n-2%n-1, if i = n - 1, 
The halving group of P, denoted by H(P), is the image of Aut(P) under .. 
Observe that the group H(P)= h.1,...,.n-1i has the following diagram 
.n-2 
pn-2 • 
.0.1.2.n-4 .n-3 
s (3.2) 
... p1 p2 
pn-3 
pn-2 • .n-1 pn-1 
where s = pn-1 if pn-1 is odd, otherwise s = . We denote by H(P) the coset inci-
 2 
dence system . H(P), (Hi), where Hi is the subgroup of H(P) generated 
i.{0,...,n-1} 
by {.j : j 6In Theorem 3.1 we show that the group H(P) satisfes the intersection 
= i}. property and in Proposition 3.2 we show that the corresponding incidence H(P) is fag-transitive. We conclude the section with Corollary 3.3 which states that H(P) is infacta thin, chamber-transitive coset geometry, i.e.a regularhypertope. 
The halving operation has been used before in the context of regular polyhedra of type {q, 4} (see[17]and[18, Section 7B]) and the resulting incidence systemisa regular poly­hedron of type {q, q}. 
The operation described above doubles the fundamental region of Aut(P) by gluing together the base fag . and the fag .n-1 . 
As an example we explore the halving operation applied to the cubic tessellation {4, 3, 4}. The elements of type 0 and 1 of the resulting incidence system are the vertices and edges of the {4, 3, 4}, respectively. The elements of type 2 are half of the cubes and the elements of type 3 arethe other half. Thisisthe constructionofthe infnitehypertope describedin[11, Example 2.5] and can alsobe seen asa semi-regular polytope (see[19, Section 3]). 
It is easy to see that H(P) has index 2 in Aut(P) if and onlyif the setoffacetsof P is bipartite. This is only possible if pn-1 is even. If this is the case, then the elements of type i, for i .{0,...,n - 3} are thefacesof rank i of P. The elements of type n - 2 are half of thefacetsof P (those belonging to the same partition as the basefacet) and the elements of type n - 1 aretheotherhalfofthefacets,namely,thoseinthesame partitionasthefacet of .n-1 . 
In the remainder of the section we let P be a fxed regular n-polytope with a base fag ., the automorphism group Aut(P)= h%0,...%n-1i and H = h.0,...,.n-1i the halving group of P. For i, j .{0,...,n - 1} we let Hi and Hi,j be the groups h.k : k 6
= ii and h.k : k 6. {i, j}i, respectively. Finally, by H(P) we denote the incidence 
.
 
system . H, (Hi)i.{0,...,n-1} and by .i the residue of H(P) induced by Hi, that is 
.
 
.i =. Hi, (Hi,j)j.{0,...,n-1}\{i} . 
Theorem 3.1. Let n > 3 and P be a regular, non-degenerate n-polytope of type {p1,..., pn-1}. Then the halving group H(P) is a C-group. 
Proof. The strategy of this proof is to use Proposition 2.3. To do so, we proceed by in­duction over n. Let .= {F-1,...,Fn} be the base fag of P. Let F 0 be thefacet of 
n-1 .n-1 
. 
If n =3, we need to prove that the group H0 = h.1,.2i = h%1,%2%1%1i is a C-group. However, this group is a subgroup of the automorphism group of the polygonal section F3/F1 isomorphic to the dihedral group Ds. The groups H1 = h.0,.2i and H2 = h.0,.1i are the automorphism groups of the polygonal sections F 20 /F-1 and F2/F-1, respectively. It follows that they are C-groups. 
To fnish our base case we only need to show 
h.0,.1i.h.0,.2i = h.0i , (3.3) 
h.0,.1i.h.1,.2i = h.1i , (3.4) 
h.0,.2i.h.1,.2i = h.2i . (3.5) 
A. Montero and A. Ivi´
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To prove(3.3), just observe that h.0i = StabP ({F1,F2}). Let . .h.0,.1i.h.0,.2i. Since . .h.0,.1i, . fxes F2. Similarly, since . .h.0,.2i, . must fx F 0 . This implies 
2
that . fxes F1, since this is the only 1-face of P incident to both F2 and F 20 . Therefore, . . StabP ({F1,F2})= h.0i. The other inclusion is obvious. 
Similarly, we have that h.0,.1i.h.1,.2i. StabP ({F0,F1}). This follows from thefact that the group h.0,.1i fxes F2 and the group h.1,.2i fxes F0. Then, h.0,.1i. h.1,.2i.h.1i. Again, the other inclusion is obvious. The proof of(3.4)follows from the same argumentbut now with respect to the fag .2 = {F0,F1,F 20 } of P. This completes the base case. 
Assume that the halving group H(F) of every non-degenerate regular polytope F of rank r with 3 6 r<n is a C-group. 
Observe that the groups Hn-1 = h.0,...,.n-2i and Hn-2 = h.0,...,.n-3,.n-1i are the automorphism groups of the sections Fn-1/F-1 and F n0-1/F-1, respectively. Hence, these groups are C-groups (see[18, Proposition 2B9]). 
Observe that if i .{0, 1,...,n - 3} then Hi = H- × H+ where H- = h.j : j<ii 
ii i 
and H+ = h.j : i<ji. Note that H- is just the automorphism of the section Fi/F-1, 
ii 
hence a C-group. If i<n - 3 then H+ is the halving group of the section Fn/Fi, which is 
i 
a C-groupby the inductivehypothesis. If i = n - 3, then H+ is isomorphictoadihedral 
i 
group Ds. In anycase, it follows from Lemma 2.4 that Hi is a C-group. In order to use Proposition 2.3, we need to prove that for every pair i, j .{0,...,n - 1}, with i<j, the equality Hi . Hj = Hi,j (3.6) 
holds. We proceed in a similar way as in rank 3. If {i, j} = {n - 1,n - 2}, then observe that Hi = Hn-2 fxes Fn-1 and Hj = Hn-1 fxes F n0-1. This implies that an element . . Hn-2 . Hn-1 must fx Fn-2. Thus . . StabP ({Fn-2,Fn-1})= h.0,...,.n-3i = Hi,j . The other inclusion is obvious. 
If j .{n - 1,n - 2} and i 6 n - 3 then(3.6)follows from thefact thatHj is a string C-group. 
Assume that 0 6 i<j 6 n - 3. Let F be the section Fj /F-1 of P. Let . . Hi . Hj . Observe that Hj = H- × H+ and that Aut(F)= H- . Let . . H- and ß . H+ be such 
jj jjj 
that . = .ß. Note that ß fxes theface Fi of P and since . . Hi it follows that . must fx Fi. Since H- is a string C-group, it follows that . .h.0,...,.i-1,.i+1,...,.j-1i. 
j 
Then 
. .h.0,...,.i-1,.i+1,...,.j-1i×h.j+1,...,.n-1i = Hi,j. 
The other inclusion is obvious. 
The halving group of a regular polytope P isa C-group with Coxeter diagram(3.2). The groups generatedbyinvolutions with this diagram are called tail-triangle groups (even when s =2). In[19,20]Monson and Schulte show that whena tail-triangle groupisa C-group, it is the automorphism group of an alternating semi-regular polytope. We de­note by S(P) the semi-regular polytope obtained by the halving operation on P. The polytope S(P) has two orbits of isomorphic regular facets, namely the right cosets of h.0,...,.n-3,.n-2i and h.0,...,.n-3,.n-1i. These base facets are incident witha reg­ular polytope R of rank n - 2. Infact,every fagof R can be extended to a fag of S(P) in two different ways. Moreover, anyfag of S(P) belongs to the orbit of one of these two fags. The automorphisms of S(P) can now be used to show fag transitivity of H(P). 
Proposition 3.2. The incidence system H(P) associated with a non-degenerate regular polytope P is fag-transitive. 
Proof. Let J .{0,...,n - 1} and let F = {Hih : i . J} for some h . H(P) be a fag of H(P) of type J. If |J .{n - 2,n - 1}| 6 1, then F is a chain of S(P) of type J. By [19, Lemma 4.5b], the group H(P) is transitive on the chains of this type. 
If {n - 2,n - 1}. J, then F =. . .n-1, where . is the chain of S(P) of type J0 = J \{n - 1} whose faces are contained in F . Again,[19, Lemma 4.5b] implies that H(P) is transitive in chains of type J0 . Finally, observe that if ß . H(P), then (.ß)n-1 = (.n-1)ß. It follows that H(P) is also transitive on fags of type J. 
Corollary 3.3. Let P be a non-degenerate, regular n-polytope and I = {0,...,n - 1}. Let H(P) be the halving group of P. Then the incidence system H(P) = .(H(P), (Hi)i.I ) is a regular hypertope suchthat AutI (H(P)) = H(P). 
The assumption that P is non-degenerateisvery important. Whenthe halving operation is applied on the dual of the 4-hemicubethe resulting incidence systemisnotahypertope (see[11, Example 3.3]). 
Theorem 2.5 in[20]implies that the semi-regular polytopeS(P) is regular because the associated C-group admits a group automorphism interchanging the generators .n-1 and .n-2 (this automorphism is given by conjugation by %n-1). The polytope S(P) is infact isomorphic to P. 
4 Locally spherical hypertopes from generalised cubes. 
Aspherical hypertope isauniversalregularhypertope whoseCoxeter diagramisa union of diagrams of fnite irreducible Coxeter groups. This defnition is slightly different from that in[11].A locally spherical hypertope isaregularhypertope whoseall proper residues are sphericalhypertopes.An irreducibleregularhypertopeisof euclidean (resp. spherical) type if the type-preserving automorphism group of its universal regular cover is an affne (resp. fnite irreducible) Coxeter group. Similarly, an irreducible locally sphericalhyper­tope is of hyperbolic type if the type-preserving automorphism group of its universal cover isa compacthyperbolicCoxeter group.Itiswellknownthat compacthyperbolicCoxeter groups exist only in ranks 3, 4 and 5. 
In[11]the authorsshowthata locally sphericalhypertopehastobeof spherical, eu­
clidean orhyperbolic type. The complete listof the Coxeter diagramsof these groups can be found in[11,Tables1 and 2]. Whereas the frst two classes are well understood, not muchisknownaboutthehyperbolictype.In particular,the authorswerenot successfulin producingafniteexampleof rank 5 locally spherical properhypertopeofthistype.Inwhat follows we will use the halving operation on a certain class of polytopes frst described by Danzerin[5](see also[18, Theorem 8D2]) andin particular we produce anexampleof fnite rank 5 proper regularhypertopeofhyperbolic type. 
We briefy review Danzer’s construction of generalised cubes. 
Let K bea regular fnite non-degenerate n-polytope withvertex set V = {v1,...,vm}. Consider the set 
m
Y 
2V = {0, 1} = {—): xj .{0, 1}} . x=(x1,...,xmj=1 
A. Montero and A. Ivi´
cWeiss: Locally spherical hypertopesfromgeneralised cubes 
Given an i-face F of K and x—. 2V defne the sets 
 
F (—x)= y=(y1,...,ym) . 2V : yj 6. 
—= xj if vj 6 F 
Then the polytope 2K is the set 
 
{Ø} . 2V . F (—x) : F .K, x—. 2V 
orderedby inclusion. The improperfaceof rank -1 of 2K is Ø and if i > 0 the i-faces of 2K are the sets F (—x) for F a certain (i - 1)-face of K and some x—. 2V . Note that F-1(—x) = {x—} for every x—. 2V , hence 2K has 2|V | vertices. 
If K isaregularof type {p1,...,pn-2} then 2K isaregular polytopeoftype {4,p1,..., pn-2}.Infact, all thevertex fguresof 2K are isomorphic to K. The polytope 2K is called a generalised cubesince when K is the (n - 1)-simplex, the polytope 2K is isomorphic to the n-cube. 
2K * 
* 
For our purposes it is convenient to denote by^2K the polytope , so that 2^K is a regular polytope of type {p1,...,pn-2, 4} whosefacets are isomorphic to K. 
The automorphism group of ^2K is isomorphic to Zm oAut(K), where m denotes the 
2 
number of facets of K and the action of Aut(K) on Zm is given by permuting coordi­
2 
nates in the natural way. In particular, the size of this group is 2m ×| Aut(K)| (see[18, Theorem 2C5] and[23]). 
Remark 4.1. In[23]Pellicer generalises Danzer’s constructionof2K. Given a fnite non-degenerate regular (n - 1)-polytope K of type {p1,...,pn-2} and s . N, Pellicer’s con­struction gives as a result an n-polytope Ps of type {p1,...pn-2, 2s}. If s =2, the poly­tope P2 is isomorphic to ^2K. However, when our construction is applied to the polytopes Ps for s > 3, the resultinghypertopes are not locally spherical and therefore not included in this paper. 
Now we discuss the locally spherical hypertopes resulting from applying the halv­ing operation to the polytopes obtained from Danzer’s construction. Since ^2K is of type {p1,...pn-2, 4}, thehypertope H(^2K) has the following Coxeter diagram: 
.n-2 pn-2 
.0.1.2.n-4 .n
-3 
... 
• 
p1 p2 pn-3 
pn-2 • .n-1 
We naturally extend the Schlafi symbol and say that ¨H(^2K) is of type {pn-1,..., 
pn-2 
pn-3, pn-2 }. 
In rank 3 the polytope ^2{p} is obtained by applying the construction on a regular poly­gon {p} and the inducedhypertopeisinfacta self-dual polyhedronof type {p, p}. This polyhedron has 2p-1 vertices, 2p-2p edges and 2p-1 faces and it is a map on a surface of genus 2p-3(p - 4) + 1.For p =3 the resultinghypertopeisa spherical polyhedron {3, 3}, 
i.e. the tetrahedron. When p =4 the polytope ^2{4} is the toroid {4, 4}(4,0) and the induced hypertope is also of euclidean type, more precisely, it is the toroid{4, 4}(2,2). 
To obtain locallysphericalhypertopes in rank 4, K must be of type {p, 3} with p = 3, 4, 5. The resultinghypertopes are of spherical, euclidean, andhyperbolic type, respec­tively. If p =3, thehypertope H(2^K) is the universalhypertope of Coxeter diagram D4 
3 
and type {3, 3}. When p =4 the polytope 2^K is the toroid {4, 3, 4}and H(^2K) is a 
(4,0,0) 
 
toroidalhypertope describedbyEnsin[6, Theorem4.3], whichwe denoteby 4, 3 . 
3 (4,0,0)
The automorphism groupofthishypertopehasCoxeter diagram B~3. If p =5, the resulting 
 
3 
hypertope is of type 5, 3 with automorphism group of size 211 × 120 = 245, 760. This exampleis different from anyof theexamples listedin[11]. 
In rank 5 the polytope K must be of type {p, 3, 3} with p =3, 4, 5, the resultinghyper­topes are of spherical, euclidean andhyperbolic type, respectively. If p =3 then thehy-
 
3 
pertopesis the universal sphericalhypertopeof type 3, 3, 3 , i.e. the universalhypertope of Coxeter diagram D5. If p =4 the polytope ^2K is the regular toroid {4, 3, 3, 4}
(4,0,0,0). The inducedhypertopeisof euclidean type, hencea toroidalhypertope which we denote 
 
3 
by 4, 3, . The Coxeter diagram of its automorphism group is B~4. For p =5 
3 (4,0,0,0)
the regular polytope ^2K is constructed from the 120-cell. Thehypertope H(^2K) is of type 
 
5, 3, 3 and its automorphism group has size 2119 × 14400. It is not surprising that the 
3 
authorsof[11]could not fnd thisexample usinga computational approach. 
For rankn > 6 we can only obtain locally sphericalhypertopes from our construction if K is the (n-1)-simplex {3n-2} or the (n-1)-cube {4, 3n-3}. The polytope ^2K is the n-cross-polytope {3n-2 , 4} or the toroid {4, 3n-3 , 4}(4,0,...0),respectively. In the former case 
3 
the resultinghypertope is the universal sphericalhypertope of type {3n-3 , 3} associated with the Coxeter diagram Dn whileinthe latteritisa toroidalhypertope associated with 
3 
the Coxeter diagram B~n-1 which we denote by {4, 3n-4 , 3}(4,0,...0). 
ORCID iDs 
Antonio Montero 
https://orcid.org/0000-0002-3293-8517 
Asia Ivi´cWeiss 
https://orcid.org/0000-0003-4937-2246 
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