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Many projectively unique polytopes

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Abstract

We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to \(96\). This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its \(f\)-vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard in the sixties. Moreover, our methods naturally lead to several interesting classes of projectively unique polytopes, among them projectively unique polytopes inscribed to the sphere. The proofs rely on a novel construction technique for polytopes based on solving Cauchy problems for discrete conjugate nets in \(S^d\), a new Alexandrov–van Heijenoort Theorem for manifolds with boundary and a generalization of Lawrence’s extension technique for point configurations.

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Notes

  1. In geology/sedimentology, “cross-bedding” refers to horizontal structures that are internally composed from inclined layers—says Wikipedia.

  2. Explicit calculations of this and the following section were performed using sagemath, Ver. 5.10.

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Acknowledgments

We are grateful to Alexander Bobenko for background information on Q-nets, to Igor Pak for valuable discussions, to Francisco Santos for particularly helpful suggestions, and to Miriam Schlöter for some of the figures in this paper. The first author thanks the Hebrew University Jerusalem, whose hospitality he enjoyed on several occasions while working on this paper.

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Correspondence to Günter M. Ziegler.

Additional information

This work was supported by the DFG within the research training group “Methods for Discrete Structures” (GRK1408) and by the Romanian NASR, CNCS-UEFISCDI, project PN-II-ID-PCE-2011-3-0533. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels and the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics”.

A: Appendix

A: Appendix

1.1 A.1 Convex position of polytopal complexes

In this section, we establish polyhedral analogues of the Alexandrov–van Heijenoort theorem, a classical result in the theory of convex hypersurfaces.

Theorem 6.1.1

([17, Main Theorem]) Let \(M\) be an immersed topological \((d-1)\)-manifold without boundary in \({\mathbb {R}}^d,\, d\ge 3\), such that

  1. (a)

    \(M\) is complete with respect to the metric induced on \(M\) by the immersion,

  2. (b)

    \(M\) is connected,

  3. (c)

    \(M\) is locally convex at each point (that is, every point \(x\) of \(M\) has a neighborhood, w.r.t. the topology induced by the immersion, in which \(M\) coincides with the boundary of some convex body \(K_x\)), and

  4. (d)

    \(M\) is strictly convex in at least one point (that is, for at least one \(x\in M\), there exists a hyperplane \(H\) intersecting the convex body \(K_x\) only in \(x\)).

Then \(M\) is embedded, and it is the boundary of a convex body.

This remarkable theorem is due to Alexandrov [5] in the case of surfaces. Alexandrov did not state it explicitly (his motivation was to prove far stronger results on intrinsic metrics of surfaces), and his proof does not extend to higher dimensions. Van Heijenoort also proved Theorem 6.1.1 only for surfaces, but his proof extends to higher dimensions. Expositions of the general case of this theorem and further generalizations are available in [46] and [35]. In this section, we adapt Theorem 6.1.1 to polytopal manifolds with boundary. We start off by introducing the notion of immersed polytopal complexes.

Definition 6.1.2

(Precomplexes) Let \(C\) denote an abstract polytopal complex, and let \(f\) denote an immersion of \(C\) into \({\mathbb {R}}^d\) (resp. \(S^d\)) with the property that \(f\) is an isometry on every face \(\sigma \) of \(C\). Then \(f(C)\) is called a precomplex in \({\mathbb {R}}^d\) (resp. \(S^d\)). While \(f(C)\) is not necessarily a polytopal complex, the subset \(f(\mathrm{St}(\sigma ,C))\) is a polytopal complex combinatorially equivalent to \(\mathrm{St}(\sigma ,C)\) for every face \(\sigma \) of \(C\).

A polytopal complex \(C\), abstract or geometric, is a d-manifold if for every vertex \(v\) of \(C\), \(\mathrm{St}(v,C)\) is PL-homeomorphic to a \(d\)-simplex [33]. If \(C\) is a manifold, then \(f(C)\) is called a premanifold. If \(\sigma \) is a face of \(C\), then \(f(\sigma )\) is a face of the polytopal precomplex \(f(C)\). The complexes \(\mathrm{St}(f(\sigma ),f(C))\) and \(\mathrm{Lk}(f(\sigma ),f(C))\) are defined to be the polytopal complexes \(f(\mathrm{St}(\sigma ,C))\) and \(\mathrm{Lk}(f(\sigma ), f(\mathrm{St}(\sigma ,C)))\) respectively.

Definition 6.1.3

(Gluing two precomplexes along a common subcomplex) Consider two precomplexes \(f(C)\), \(g(C')\), and assume there are subcomplexes \(D\), \(D'\) of \(C\), \(C'\) respectively with \(f(D)=g(D')\) such that for every vertex \(v\) of \(f(D)=g(D')\), \(\mathrm{St}(v,f(C))\cup \mathrm{St}(v,g(C'))\) is a polytopal complex. Let \(\Sigma \) be the abstract polytopal complex given by identifying \(C\) and \(C'\) along the map \(g^{-1}\circ f:D\rightarrow D'\), and let \(s\) denote the immersion of \(\Sigma \) defined as

$$\begin{aligned} s(x):=\Big \{\begin{array}{l@{\quad }l} f(x) &{}\ \text {for }x\in C, \\ g(x)&{}\ \text {for }x\in C'. \end{array} \end{aligned}$$

Then the gluing of \(f(C)\) and \(g(C')\) at \(f(D)=g(D')\), denoted by \(f(C)\sqcup _{f(D)} g(C')\), is the polytopal precomplex obtained as the image of \(\Sigma \) under \(s\).

For the rest of the section, we will use the term halfspace in \(S^d\) synonymously for a hemisphere in \(S^d\).

Definition 6.1.4

(Locally convex position and convex position for polytopal (pre-)complexes) A pure polytopal (pre-)complex \(C\) in \(S^d\) or in \({\mathbb {R}}^{d}\) is in convex position if one of the following three equivalent conditions is satisfied:

  • For every facet \(\sigma \) of \(C\), there exists a closed halfspace \(H(\sigma )\) containing \(C\) such that \(\partial H(\sigma )\) contains the vertices of \(\sigma \) but no other vertices of \(C\). We say that such a halfspace \(H(\sigma )\) exposes \(\sigma \) in \(C\).

  • Every facet is exposed by some linear functional, i.e. there exists, for every facet \(\sigma \) of \(C\), a vector \(\vec {n}(\sigma )\) such that the points of \(\sigma \) maximize the linear functional \(\langle \vec {n},x\rangle \) among all points \(x\in C\). For \(C\subset S^d\), we additionally demand \(\langle \vec {n}(\sigma ),x\rangle =0\) for all \(x\) in \(\sigma \).

  • \(C\) is a subcomplex of the boundary complex of a convex polytope.

Likewise, a polytopal (pre-)complex \(C\) in \(S^d\) or \({\mathbb {R}}^{d}\) is in locally convex position if for every vertex \(v\) of \(C\), the link \(\mathrm{Lk}(v,C)\), seen as a subcomplex in the \((d-1)\)-sphere \({N}^1_v {\mathbb {R}}^d\) resp. \({N}^1_v S^d\), is in convex position. As \(\mathrm{Lk}(v,C)\) is in convex position if and only if \(\mathrm{St}(v,C)\) is in convex position, \(C\) is in locally convex position if and only if \(\mathrm{St}(v,C)\) is in convex position for every vertex \(v\) of \(C\).

It is obvious that “convex position” implies “locally convex position.” The Alexandrov–van Heijenoort theorem describes conditions under which this observation can be reversed. We start with a direct analogue of the Theorem 6.1.1 for precomplexes. Notice that a precomplex without boundary in locally convex position is locally convex at every point and strictly convex at every vertex in the sense of Theorem 6.1.1.

Theorem 6.1.5

(AvH for closed precomplexes) Let \(C\) be a \((d-1)\)-dimensional connected closed polytopal premanifold in \({\mathbb {R}}^d\) or \(S^d\), \(d\ge 3\), in locally convex position. Then \(C\) is in convex position.

Proof

In the euclidean case, the metric induced on \(C\) is complete because \(C\) is finite, and \(C\) is locally convex at each point since \(C\) is in locally convex position. Furthermore, \(C\) is strictly convex at every vertex of \(C\). Thus by Theorem 6.1.1, \(C\) is the boundary of a convex polytope. Since every facet of \(C\) is exposed by a linear functional, the boundary complex of this polytope coincides with \(C\).

For the spherical case, let \(v\) be a vertex of \(C\), and let \(P\) be the polyhedron in \(S^d\) that is obtained by intersecting the halfspaces exposing the facets of \(\mathrm{St}(v,C)\). Let \(H\) denote a closed halfspace containing \(P\), chosen so that \(\partial H\cap \mathrm{St}(v,C)=v\). As \(C\) is polytopal, the complex \(C\) contains at least one vertex \(w\) in the interior of \(H\). Consider any central projection \(\zeta \) mapping \(\mathrm{int}H\) to \({\mathbb {R}}^d\). Then \(|\zeta (C)|\) is the boundary of a convex polyhedron \(K\) in \({\mathbb {R}}^d\) by Theorem 6.1.1. In particular \(K\subseteq \zeta (P)\). Since \(K\) is pointed at the vertex \(\zeta (w)\), \(K\) contains no line, and consequently, \(\mathrm{cl}\zeta ^{-1}(K)\) contains no antipodal points. Thus

$$\begin{aligned} \mathrm{cl}\zeta ^{-1}(K)=\zeta ^{-1}(K)\cup v\subsetneq P \end{aligned}$$

is a polytope in \(S^d\). Since every facet of \(C\) is exposed by a halfspace, the boundary complex of \(\zeta ^{-1}(K)\) is \(C\), as desired. \(\square \)

Example 6.1.6

A polytopal premanifold \(C\) is called simple if for every vertex \(v\) of \(C\), \(\mathrm{Lk}(v,C)\) is a simplex. Consider now any simple, closed and connected \(k\)-premanifold \(C\) in \({\mathbb {R}}^d\), where \(d> k\ge 2\). Since \(C\) is simple and connected, it is contained in some affine \((k+1)\)-dimensional subspace of \({\mathbb {R}}^d\). Since \(C\) is simple, it is either in locally convex position, or locally flat (i.e. locally isometric to \({\mathbb {R}}^k\)). Since \(C\) is furthermore compact, only the former is possible. To sum up, \(C\) is a \(k\)-dimensional premanifold that is closed, connected and in locally convex position in some \((k+1)\)-dimensional affine subspace. Hence \(C\) is in convex position by Theorem 6.1.5 (cf. [16, Sec. 11.1, Pr. 7]).

Definition 6.1.7

(Fattened boundary) Let \(C\) be a polytopal \(d\)-manifold, and let \(B\) be a connected component of its boundary. The fattened boundary \(\mathrm{fat}(B,C)\) of \(C\) at \(B\) is the minimal subcomplex of \(C\) containing all facets of \(C\) that intersect \(B\) in a \((d-1)\)-face.

Lemma 6.1.8

(Gluing lemma) Let \(C\), \(C'\) denote two connected polytopal \((d-1)\)-manifolds with boundary in \(S^d\) or \({\mathbb {R}}^d,\, d\ge 2 \) with \(B:=\partial C= \partial C'\). Assume that \(C\) and \(C'\cup \mathrm{fat}(B,C)\) are in convex position. Then \(C\cup C'\) is the boundary complex of a convex polytope.

Proof

We proceed by induction on the dimension. First, consider the case \(d=2\), whose treatment differs from the case \(d> 2\) since Theorem 6.1.1 is not applicable. We use the language of curvature of polygonal curves, cf. [43]. If \(C\), \(C'\) are in \(S^2\), use a central projection to transfer \(C\) and \(C'\) to complexes in convex position in \({\mathbb {R}}^2\). If there are two curves \(C\) and \(C'\) in convex position in \({\mathbb {R}}^2\) such that \(C'\cup \mathrm{fat}(B,C)\) is in convex position, then \(C\sqcup _{\mathrm{fat}(B,C)} C'\) is a 1-dimensional premanifold whose curvature never changes sign, and which is of total curvature less than \(4\pi \) since the total curvature of \(C'\cup \mathrm{fat}(B,C)\) is smaller or equal to \(2\pi \), and \(C\) has total curvature less than \(2\pi \). Since the turning number of a closed planar curve is a positive integer multiple of \(2\pi \), the total curvature of \(C\sqcup _{\mathrm{fat}(B,C)} C'\) is \(2\pi \). By Fenchel’s Theorem [13], \(C\sqcup _{\mathrm{fat}(B,C)} C'\) is the boundary of a planar convex body. Since every facet is exposed, the boundary complex of this convex body must coincide with \(C\sqcup _{\mathrm{fat}(B,C)} C'=C\cup C'\).

We proceed to prove the lemma for dimension \(d> 2\). If \(v\) is a vertex of \(B\), then \(\mathrm{Lk}(v,C\sqcup _{\mathrm{fat}(B,C)} C')\) is obtained by gluing the two complexes \(\mathrm{Lk}(v,C)\) and \(\mathrm{Lk}(v,C')\cup \mathrm{Lk}(v,\mathrm{fat}(B,C))\) along \(\mathrm{fat}(B,C)\). Each of these is of codimension 1 and in convex position, so the resulting complex is a polytopal sphere in convex position by induction on the dimension. In particular, \(C\sqcup _{\mathrm{fat}(B,C)} C'\) is a premanifold in locally convex position. Thus by Theorem 6.1.5, \(C\sqcup _{\mathrm{fat}(B,C)} C'=C\cup C'\) is in convex position. \(\square \)

We will apply Theorem 6.1.5 in the following version for manifolds with boundary:

Theorem 6.1.9

Let \(C\) be a polytopal connected \((d-1)\)-dimensional (pre)-manifold in locally convex position in \({\mathbb {R}}^d\) or in \(S^d\) with \(d\ge 3\), and assume that for all boundary components \(B_i\) of \(\partial C\), their fattenings \(\mathrm{fat}(B_i,C)\) are (each on its own) in convex position. Then \(C\) is in convex position.

Proof

Consider any boundary component \(B\) of \(C\) and the boundary complex \(\partial \mathrm{conv}\mathrm{fat}(B,C)\) of the convex hull of \(\mathrm{fat}(B,C)\), the fattened boundary of \(C\) at \(B\). The subcomplex \(B\) decomposes \(\partial \mathrm{conv}\mathrm{fat}(B,C)\) into two components, by the (polyhedral) Jordan–Brouwer Theorem. Consider the component \(A\) that does not contain the fattened boundary \(\mathrm{fat}(B,C)\) of \(C\). The \((d-1)\)-complex \(A\cup \mathrm{fat}(B,C)\subseteq \partial \mathrm{conv}\mathrm{fat}(B,C)\) is in convex position, and by Lemma 6.1.8, the result \(A\sqcup _{\mathrm{fat}(B,C)} C\) of gluing \(C\) and \(A\) at \(B\) is a premanifold in locally convex position.

Repeating this with all boundary components yields a polytopal premanifold without boundary in locally convex position. Thus it is the boundary of a convex polytope, by Theorem 6.1.5. Since \(C\) is still a subcomplex of the boundary of the constructed convex polytope, \(C\) is in convex position. \(\square \)

1.2 A.2 Duality, reciprocals, convex liftings and cross-bedding cubical tori

In this section, we outline an elegant alternative proof of Main Theorem I. The punchline is that convex position of the extension (Theorem 4.0.7) is an automatic corollary of the existence of the extension. To show this we make use of a relation between reciprocals (or orthogonal duals) and convex liftings of polytopal complexes based on the Maxwell–Cremona correspondence [11, 23]. The arguments in this section are only sketched, and there are some substantial disadvantages compared to the approach detailed in the main part of this paper (based on the Alexandrov–van Heijenoort Theorem) that we will detail on at the end.

The section has two parts: In Sect. A.2.1, we sketch the necessary notions and methods for duals, reciprocals and convex liftings. In Sect. A.2.2, we apply these ideas to our cross-bedding cubical tori.

1.2.1 A.2.1 Duals, reciprocals and liftings to convex position

The following summary of basic notions and results concerning reciprocal complexes and their convex liftings loosely follows Rybnikov [34]. For details and an intuitive explanations of the following results, we refer the reader to [6, 9, 34]; more detailed references are collected at the end of this section. All polytopal manifolds in this section are manifolds with boundary.

Definition 6.2.1

(Duality) Let \(C\) be a polytopal \(d\)-manifold. A complex \(D\) is dual to \(C\) if there is an injective map \(\delta :D\rightarrow C\) such that

  • \(k\)-dimensional faces of \(D\) map to \((d-k)\)-dimensional faces of \(C\),

  • \(\delta \) is a bijection between \(D\) and those faces of \(C\) not in \(\partial C\), and

  • if \(\tau , \sigma \) are faces of \(D\), \(\tau \subset \sigma \), then \(\delta (\sigma )\subset \delta (\tau )\).

For a manifold in convex position, there is a natural dual: If \(C\) is a polytopal \(d\)-manifold in convex position in \(S^{d+1}\), denote by \((\mathrm{conv}C)^*\) the polar dual of the convex polytope \(\mathrm{conv}C\), i.e.

$$\begin{aligned} (\mathrm{conv}C)^*:=\{x\in S^{d+1}: \langle x,y\rangle \le 0\ \text {for all}\ y\in \mathrm{conv}C\}. \end{aligned}$$

Then the polar \(C^*\) to \(C\) is the subcomplex of \(\partial (\mathrm{conv}C)^*\) consisting of faces of \(\partial (\mathrm{conv}C)^*\) corresponding to faces of \(C\) not in \(\partial C\).

Definition 6.2.2

(Reciprocity) Two subspaces \(V, W\subset S^d\) with \( W \cap V=\{x, -x\}\) for some \(x\in S^d\) are reciprocal if the linear subspaces \({T}_x V\) and \({T}_x W\) are orthogonal complements in \({T}_x S^d\cong {\mathbb {R}}^{d}\) (In particular, this implies that \(\dim V+\dim W=d\)). A dual \(D\subset S^d\) to a polytopal \(d\)-manifold \(C\) in \(S^d\) is a reciprocal of \(C\) if for every face \(\sigma \) of \(D\), the subspaces \(\mathrm{sp}(\sigma )\) and \(\mathrm{sp}(\delta (\sigma ))\) in \(S^d\) are reciprocal.

Observation 6.2.3

Let \(V, W, V'\) and \(W'\) be subspaces of \(S^d\). Assume that \(V\) is reciprocal to \(W\), that \(V'\) is reciprocal to \(W'\), and that \(\dim \mathrm{sp}(V\cup V')= \dim V +\dim V'=d-\dim W\cap W'\). Then the subspaces \(\mathrm{sp}(V\cup V')\) and \(W\cap W'\) are reciprocal.

Hence, checking reciprocity may be restricted to edges of \(D\):

Lemma 6.2.4

Let \(C\) be a polytopal \(d\)-manifold in \(S^d\), and let \(D\) in \(S^d\) be a dual to \(C\). Then \(D\) is a reciprocal of \(C\) if and only if for every edge \(e\) of \(D\), the subspaces \(\mathrm{sp}(e)\) and \(\mathrm{sp}(\delta (e))\) are reciprocal. \(\square \)

If \(\sigma \), \(\tau \) are adjacent facets of a polytopal \(d\)-manifold \(C\) in \(S^d\), then let us denote by \(\mathrm{n}^\sigma _\tau \) the normal to the face \(\sigma \cap \tau \) directed towards \(\sigma \), i.e. \(\mathrm{n}^\sigma _\tau \) is the midpoint of the hemisphere that contains \(\sigma \), but does not intersect \(\mathrm{int}\tau \). With this, a reciprocal \(D\) to a polytopal manifold \(C\) in \(S^d\) is orientation preserving if and only if for every pair of vertices \(a\), \(b\) of \(D\) that are connected by an edge, we have \(\langle \mathrm{n}^{\delta (a)}_{\delta (b)}, a-b \rangle >0.\)

Proposition 6.2.5

Let \(C\) be a polytopal \(d\)-manifold in convex position in \(S^{d+1}\). If the orthogonal projection \(\mathrm{p} :S^{d+1}\rightarrow S^d_{\mathrm{eq}}\) is well-defined and injective on \(C\) and on its polar \(C^*\), then \(\mathrm{p} (C^*)\) is an orientation preserving reciprocal for \(\mathrm{p} (C)\). \(\square \)

Proposition 6.2.5 motivates the notion of liftings of polytopal complexes.

Definition 6.2.6

(Liftings and convex liftings) Let \(C\) be a polytopal complex in \(S^d_{\mathrm{eq}}\). A complex \(\widehat{C}\) in \(S^{d+1}\) is a lifting of \(C\) if the orthogonal projection \(\mathrm{p} \) is well-defined and injective on \(\widehat{C}\) and \(\mathrm{p} (\widehat{C})=C\). The complex \(\widehat{C}\) is a convex lifting, or lifting to convex position, of \(C\) if the lifting \(\widehat{C}\) of \(C\) is in convex position.

Theorem 6.2.7

Let \(C\) be a polytopal \(d\)-manifold in \(S^d_{\mathrm{eq}}\) with \(H_1(C,\mathbb {Z}_2)=0\). Then \(C\) admits an orientation preserving reciprocal if and only if it admits a convex lifting to \(S^{d+1}\).

For our intended application, we need a generalization applicable to manifolds with general topology:

Theorem 6.2.8

Let \(B\) be a polytopal \(d\)-manifold in convex position in \(S^{d+1}\) on which \(\mathrm{p} \) is well-defined and injective, and let \(C\) be a polytopal \(d\)-manifold in \(S^d_{\mathrm{eq}}\) so that \(\mathrm{p} (B)\) is a subcomplex of \(C\). Assume that

  1. (a)

    the inclusion \(\mathrm{p} (B)\rightarrow C\) induces a surjection \(H_1(\mathrm{p} (B),\mathbb {Z}_2)\rightarrow H_1(C,\mathbb {Z}_2)\),

  2. (b)

    \(C\) admits a reciprocal \(D\), and

  3. (c)

    the natural combinatorial isomorphism \(\mathrm{p} (B^*)\rightarrow \delta ^{-1}(\mathrm{p} (B))\) is geometrically realized by the identity on \(S^d_{\mathrm{eq}}\).

Then \(C\) admits a convex lifting \(\widehat{C}\) such that the subcomplex of \(\widehat{C}\) that projects to \(\mathrm{p} (B)\) coincides with \(B\).

A few words on the proof: Classically, the relation between reciprocal complexes and liftings was formulated for complexes in euclidean spaces; for the euclidean plane, a version of Theorem 6.2.7 was noticed already by Maxwell [22, 23]. For expositions of the general case and Proposition 6.2.5, compare in particular [6, Thm. 1]. The analogue of Theorem 6.2.8 for the euclidean setting follows from work of Crapo and Whiteley [9, 10]; compare in particular Theorem 2.6.3 in Rybnikov’s PhD thesis [34]. A few authors also treated the spherical case directly: in particular, McMullen [24] provided a proof for the special case \(C\cong S^d_{\mathrm{eq}}\) of Theorem 6.2.8 in the spherical setting. The translation of [34, Thm. 2.6.3] to the spherical case, and hence the proof of Theorems 6.2.8 and 6.2.7, is quite straightforward; we omit the details.

1.2.2 A.2.2 Reciprocals and cross-bedding cubical tori

In this section we prove that reciprocity is a property that is naturally preserved when extending CCTs. The main theorem is the following.

Theorem 6.2.9

Assume that \(\mathrm{T}\) and \(\mathrm{S}\) are CCTs in \(S^3\) such that

  1. (a)

    \(\mathrm{T}\) is a polytopal manifold, or equivalently, \(\mathrm{T}\) is of width at least 5,

  2. (b)

    \(\mathrm{S}\) is an orientation preserving reciprocal for \(\mathrm{T}\), and

  3. (c)

    both \(\mathrm{T}\) and \(\mathrm{S}\) admit elementary extensions, say \(\mathrm{T}'\) and \(\mathrm{S}'\).

Then \(\mathrm{S}'\) is an orientation preserving reciprocal for \(\mathrm{T}'\).

Remark 6.2.10

CCTs are special instances of Q-nets, which are discrete analogues of conjugate nets, cf. [7], Sec. 1 & 2]. By following the proof we will sketch below, it is not hard to see that Theorem 6.2.9 holds in an analogous form for Q-nets of dimension at least 3. It might be interesting to further explore of the connection between Q-nets and reciprocals.

Sketch of Proof

We only treat reciprocity of the extension, orientation preservation is left to the reader. By Lemma 6.2.4, we have to prove that for any 3-cube \(W\) of \(\mathrm{T}'\) not in \(\mathrm{T}\) and for any facet \(A\) of \(\mathrm{T}\) adjacent to \(W\), the subspaces \(\mathrm{sp}(W\cap A)\) and \(\mathrm{sp}\delta ^{-1}(W\cap A)\) are reciprocal. For this, let \(B_1\), \(B_2\) denote the remaining facets of \(\mathrm{T}\) adjacent to \(W\), and let \(F_{i}\) denote the facet of \(\mathrm{T}\) adjacent to both \(A\) and \(B_i\), \(i=1,2\). Moreover, we set \(e_i:=W\cap A\cap B_i \cap F_i,\ i=1,2\).

The proof is now simple: Since \(\mathrm{S}\) is a reciprocal for \(\mathrm{T}\), the subspaces \(\mathrm{sp}(A\cap F_i)\) and \(\mathrm{sp}\delta ^{-1}(A\cap F_i)\) are reciprocal, and so are the subspaces \(\mathrm{sp}(B_i\cap F_i)\) and \(\mathrm{sp}\delta ^{-1}(B_i\cap F_i)\). Hence, the subspaces

$$\begin{aligned} \mathrm{sp}\left( \delta ^{-1}(A\cap F_i)\cup \delta ^{-1}(B_i\cap F_i)\right)&= \mathrm{sp}\delta ^{-1}(e_i) \quad \text {and}\\ \mathrm{sp}(A\cap F_i)\cap \mathrm{sp}(B_i\cap F_i)&= \mathrm{sp}(e_i) \end{aligned}$$

are reciprocal by Observation 6.2.3. Finally, invoking Observation 6.2.3 again shows reciprocity of the subspaces

$$\begin{aligned} \mathrm{sp}(e_1\cup e_2)=\mathrm{sp}(W\cap A)\quad \text {and}\quad \mathrm{sp}\delta ^{-1}(e_1)\cap \mathrm{sp}\delta ^{-1}(e_2) = \mathrm{sp}\delta ^{-1}(W\cap A). \end{aligned}$$

\(\square \)

If we combine Theorem 6.2.8, 6.2.9 and 4.0.6, we obtain the following theorem that can replace both Theorem 4.0.5 and Theorem 4.0.7 for the proof of Main Theorem I.

Theorem 6.2.11

Let \(\mathrm{T}\) be an ideal CCT of width \(k\ge 6\) in convex position in \(S^4\). Assume that its polar \(\mathrm{S}=\mathrm{T}^*\) is ideal and in convex position as well. Then there are ideal CCTs \(\mathrm{T}'\) and \(\mathrm{S}'\), of width \(k+1\) and \(k-2\) respectively, such that

  1. (a)

    both \(\mathrm{T}'\) and \(\mathrm{S}'\) are ideal,

  2. (b)

    both \(\mathrm{T}'\) and \(\mathrm{S}'\) are in convex position,

  3. (c)

    \(\mathrm{S}'\) is the polar dual to \(\mathrm{T}'\), and

  4. (d)

    \(\mathrm{R}(\mathrm{T}',[0,k])=\mathrm{T}\) and \(\mathrm{R}(\mathrm{S}',[0,k-3])=\mathrm{S}\).

Conclusion. We now have two proof strategies for Main Theorem I that can be summarized as follows:

  1. (a)

    Start with an ideal 3-CCT in convex position in \(S^4\), for instance \(\mathrm{T}^{s}[3]\). Now, use Theorem 4.0.5 to prove that the extensions of \(\mathrm{T}^{s}[3]\) exist and are ideal, and use Theorem 4.0.7 to prove that these extensions are in convex position.

  2. (b)

    Start with an ideal 6-CCT in convex position in \(S^4\) whose polar is ideal as well. For instance, one can verify that \(\mathrm{T}^{s}[6]\) is a CCT as desired. Since the polar is automatically in convex position, we can now use Theorem 6.2.11 to prove that CCTs in convex position of arbitrary width exist.

Approach (B) is arguably more intuitive and straightforward, and it avoids several tedious arguments when checking the conditions of the Alexandrov–van Heijenoort Theorem 6.1.9. However, to use it we have to start with a CCT in convex position of considerable higher width [width 6, compared to width 3 for approach (A)] and whose polar is ideal as well. This has to be verified by hand, and is much more demanding than verifying that a 3-CCT is ideal and in convex position. This is in particular relevant if one wants to construct CCTs based on different initial layers, as we will do in Sect. A.5.

1.3 A.3 Shephard’s list

Construction methods for projectively unique \(d\)-polytopes were developed by Peter McMullen in his doctoral thesis (Birmingham 1968) directed by G. C. Shephard; see [25], where McMullen writes:

“Shephard (private communication) has independently made a list, believed to be complete, of the projectively unique 4-polytopes. All of these polytopes can be constructed by the methods described here.”

If the conjecture is correct, the list of eleven projectively unique 4-polytopes. Table 1 (all of them generated by McMullen’s techniques, duplicates removed) should be complete.

Table 1 Shephard’s list of 4-dimensional projectively unique polytopes

1.4 A.4 Iterative construction of CCTs

The main results of this paper were based on an iterative construction of ideal CCTs (Sect. 3). It is natural to ask whether one can provide explicit formulas for this iteration, and indeed, a first attempt to prove Theorems I and II would try to understand these iterations in terms of explicit formulas. Since the building block of our construction is Lemma 3.1.4, this amounts to understanding the following problem.

Problem 6.4.1

Let \(Q_1\), \(Q_2\), \(Q_3\) be three quadrilaterals in some euclidean space (or in some sphere) on vertices \(\{a_1,\, a_2,\, a_3,\, a_4\}\), \(\{a_1,\, a_4,\, a_5,\, a_6\}\) and \(\{a_1,\, a_2,\, a_7,\, a_6\}\), respectively, such that the quadrilaterals do not lie in a common 2-plane. Give a formula for \(a_1\) in terms of the coordinates of the vertices \(a_i\), \(i\in \{2,\, \dots ,\, 7\}\).

It is known and not hard to see that this formula is rational [7, Sec. 2.1]. The formula is, however, rather complicated, so that it is much easier to follow an implicit approach for the iterative construction of ideal CCTs.

In this section we nevertheless give, without proof, an explicit formula (Formula 6.4.4) to compute, given an ideal 1-CCT \(\mathrm{T}\) in \(S^4_+\), its elementary extension \(\mathrm{T}'\) by solving Problem 6.4.1 in \(S^4_+\) for cases with a certain inherent symmetry coming from the symmetry of ideal CCTs. More accurately, we provide a rational formula for a map \(\mathrm{i}\) that, given two special vertices \(a,\, b\) of \(\mathrm{T}\) in layers \(0\) and 1 respectively, obtains a vertex \(c:=\mathrm{i}(a,b)\) of layer 2 of \(\mathrm{T}'\). The map \(\mathrm{i}\) is chosen in such a way that we can easily iterate it, i.e. in order to obtain a vertex \(d\) of the elementary extension \(\mathrm{T}''\) of \(\mathrm{T}'\), we simply compute \(\mathrm{i}(b,{\mathrm{r}}^{2}_{1,2} c)\) (cf. Proposition 6.4.3).Footnote 2

Remark 6.4.2

A word of caution: Formula 6.4.4 for \(\mathrm{i}(a,b)\) is also well-defined for some values of \(a,\, b\) for which the extension \(\mathrm{T}'\) of \(\mathrm{T}\) does not exist. In particular, one should be careful not to interpret the well-definedness of \(\mathrm{i}(a,b)\) as a direct proof of Theorem 4.0.5, rather the opposite: Theorem 4.0.5 proves that the extensions of ideal CCTs exist, which allows us, if we are so inclined, to use the explicit formula for \(\mathrm{i}\) to compute them. For the rest of this section we will simply ignore this problem; we shall assume the extension exists whenever we speak of an extension of a CCT.

Explicit formula for the iteration: To define \(\mathrm{i}\), choose vertices \(a\in \mathrm{R}(\mathrm{T},0)\) and \(b\in \mathrm{R}(\mathrm{T},1)\) of the ideal 1-CCT \(\mathrm{T}\) as in Fig. 14, and, to simplify the formula, such that \(\langle a,e_4 \rangle \) and \(\langle b,e_4 \rangle \) vanish.

Fig. 14
figure 14

Set-up for an explicit iterative formula; the boundary vertices and edges of the disk are in \(\mathrm{T}\), the interior faces (those containing \(c\)) are added in the extension to \(\mathrm{T}'\)

We are going to give the formula for the vertex \(c:=\mathrm{i}(a,b) \in \mathrm{R}(\mathrm{T}',2)\) as indicated in Fig. 14. Then, it is easy to compute all extensions of an ideal CCTs in \(S^4\) explicitly using iterations of \(\mathrm{i}\).

Proposition 6.4.3

Let \(\mathrm{T}\) be an ideal 1-CCT in \(S^4_+\), and assume that \(a\in \mathrm{R}(\mathrm{T},0)\) and \(b\in \mathrm{R}(\mathrm{T},1)\) are chosen as before. Let us denote by \(\mathrm{T}^{[k]},\ k\ge 0\), the \(k\)-th layer of the \(k\)-CCT extending \(\mathrm{T}\). Set \(\kappa _0:=a\), \(\kappa _1:=b\) and define

$$\begin{aligned} \kappa _{k+1}:={\mathrm{r}}^{2}_{1,2}\mathrm{i}(\kappa _{k-1},\kappa _{k}). \end{aligned}$$

Then \(\kappa _{k}\in \mathrm{T}^{[k]}\) for all \(k\). \(\square \)

Formula 6.4.4

Consider an ideal 1-CCT \(\mathrm{T}\) in \(S^4_+\). We use homogeneous coordinates for the vertices of \(\mathrm{T}\). Then, we have the desired formula for \(\mathrm{i}\):

$$\begin{aligned} \mathrm{i}(a,b)=\mu (a,b)a+(1-\mu (a,b)) \frac{{\mathrm{r}}_{1,2}{\mathrm{r}}_{3,4} b + {\mathrm{r}}_{1,2}{\mathrm{r}}_{3,4}^{-1} b}{2}, \end{aligned}$$

where the parameter \(\mu (a,b)\) is given by

$$\begin{aligned} \mu (a,b)&= \frac{\big (\mathrm{S}+\mathrm{L}_{\hat{b}}+\mathrm{D}\big )\big (3\mathrm{S}+\mathrm{L}_{\hat{b}}+\mathrm{D}\big )}{\mathrm{L}_{\hat{b}}^2+4\mathrm{L}_{\hat{a}}\mathrm{S}+2\mathrm{L}_{\hat{b}}\mathrm{D} + 2\mathrm{S}^2+4\mathrm{D}\mathrm{S}}\ \ \text {and therefore}\ \ 1-\mu (a,b)\\&= \frac{4(\mathrm{L}_{\hat{a}}-\mathrm{L}_{\hat{b}})\mathrm{S}-\mathrm{S}^2-\mathrm{D}^2}{\mathrm{L}_{\hat{b}}^2+4\mathrm{L}_{\hat{a}}\mathrm{S}+2\mathrm{L}_{\hat{b}}\mathrm{D} + 2\mathrm{S}^2+4\mathrm{D}\mathrm{S}}. \end{aligned}$$

Here \(\mathrm{S}= \langle \hat{a},\hat{b}\rangle \) denotes the scalar product of the vectors \(\hat{a}=(a_1,a_2)\) and \(\hat{b}=(b_1,b_2)\), while \(\mathrm{L}_{\hat{a}}=||\hat{a}||_2^2=a_1^2+a_2^2\) and \(\mathrm{L}_{\hat{b}}=||\hat{b}||_2^2=b_1^2+b_2^2\) denote their respective square lengths. Finally, \(\mathrm{D}\) is the signed volume of the parallelepiped spanned by \(\hat{a}\) and \(\hat{b}\), i.e., \(\mathrm{D}\) is the determinant of the matrix \(\left( \begin{array}{l@{\quad }l} a_1 &{} b_1\\ a_2 &{} b_2\end{array}\right) \). \(\square \)

Remark 6.4.5

One can conclude several interesting facts about our construction from Formula 6.4.4:

  1. (a)

    It follows directly from the existence of the extension that the term \(\mu (a,b)\) is negative if the CCT it is applied to is ideal.

  2. (b)

    We have \(\langle \hat{b},\hat{c}\rangle =\mu (a,b)\langle \hat{a},\hat{b}\rangle \), where \(\hat{c}=(c_1,c_2)\). Hence, the term \(\mathrm{S}\) in Formula 6.4.4 converges to \(0\) at an exponential rate as ideal CCTs are extended, and does not change sign since \(\mu (a,b)<0\).

  3. (c)

    If we consider iterative extensions of ideal CCTs in convex position, then as the construction progresses, the square length of vectors in the newest layer converges (by a simple compactness argument). Hence, in the notation of Formula 6.4.4, \(\mathrm{L}_{\hat{a}}-\mathrm{L}_{\hat{b}}\) tends to \(0\).

  4. (d)

    Combining the previous two observation, we see that \(\mu (a,b)\) tends to \(0\) as the extension progresses.

  5. (e)

    We can conclude from this that as we iteratively build an ideal CCT, the squared norm of the last two coordinates of the vertices in the highest layer \(i\) can be bounded above by a constant multiple of \((\frac{\sqrt{3}}{{2}})^{i}\).

Example 6.4.6

If \(\mathrm{T}^{s}[1]\), as given in Sect. 4.1, is our starting CCT for the proof of Theorem I, then we can choose

$$\begin{aligned} a&= (\sqrt{2}-1,\,1-\sqrt{2},\, 2,\, 0,\, 1)= \vartheta _0\in \mathrm{R}(\mathrm{T}^{s}[1],0)\quad \text {and}\\ b&= (-1,\,0,\, 1,\, 0,\, 1)={\mathrm{r}}^2_{1,2}\vartheta _1\in \mathrm{R}(\mathrm{T}^{s}[1],1). \end{aligned}$$

Then, \(\mu (a,b)=\tfrac{1}{23}(3-4\sqrt{2})\) and

$$\begin{aligned} \mathrm{i}(a,b)&= \big (\tfrac{1}{23}(-11+7\sqrt{2}),\,\tfrac{1}{23}(-9-11\sqrt{2}),\, \tfrac{1}{23}(16-6\sqrt{2}),\, 0 ,\, 1\big )\\&= \vartheta _2\in \mathrm{R}(\mathrm{T}^{s}[2],2). \end{aligned}$$

More generally, by setting \(\kappa _0:=\vartheta _0\), \(\kappa _1:={\mathrm{r}}^2_{1,2}\vartheta _1\), we can use the iteration procedure of Proposition 6.4.3 to inductively construct the complexes \(\mathrm{T}^{s}[n]\) (Table 2). We compute \(\kappa _i\) explicitly for \(i\le 10\); the fourth coordinate is always \(0\) and the fifth is always 1, so we omit them from the list. Furthermore we compute the value \(\lambda (\kappa _i)\) such that \(\mathrm{p} (\kappa _i)\) lies in \(\mathcal {C}_{\lambda (\kappa _i)}\). We constructed the complexes \(\mathrm{T}^{s}[n]\) towards \(\mathcal {C}_0\), so these values should decrease. In fact, by Remark 6.4.5(c) and (e), we obtain that \(\lambda (\kappa _i)=O \big ((\frac{3}{4})^{i}\big )\xrightarrow {i\rightarrow \infty } 0\).

Table 2 Coordinates for the polytopes \(\mathrm{CCTP}_4[n]\)

1.5 A.5 Many more projectively unique polytopes

In this section we construct, for any finite field extension \(F\subset {\mathbb {R}}\) over \(\mathbb {Q}\), infinitely many projectively unique polytopes in fixed dimension that are characteristic to \(F\) (Sect. A.5.1). Moreover, we construct infinitely many inscribed projectively unique polytopes in fixed dimension (Sect. A.5.2).

Compared to Theorem II, we shall not use explicit construction methods but rather rely on general results of [3] to obtain projectively unique polytopes from polytopes with low-dimensional realization space. Hence, the “fixed dimension” of the projetively unique polytopes constructed here is only implicit (although in principle computable) and in general much larger than \(69\).

We work with polytopes and point configurations in \(S^d\); coordinates of points and vertices in the upper hemisphere \(S^d_+\subset S^d\) referred to are always homogeneous coordinates.

1.5.1 A.5.1 Many projectively unique polytopes over any field

Perles not only constructed exponentially many projectively unique polytopes [25, 29], he also famously established the existence of a projectively unique polytope not realizable in any rational vector space [16, Sec. 5.5, Thm. 4]. It is known (see [3]) that his results extend to any finite field extension: For any finite field extension \(F\subset {\mathbb {R}}\) over \(\mathbb {Q}\), there is a projectively unique polytope \(P\) that is realizable in a vector space over \(F\), but not in any strict subfield \(G\) of \(F\). In this section, we prove that there is not only one such polytope, but there even are infinitely many in some fixed dimension that depends only on \(F\). The main result is the following analogue of Main Theorem II.

Theorem 6.5.1

Let \(F\subset {\mathbb {R}}\) be any finite field extension over \(\mathbb {Q}\). For any \(d \ge D=D(F)\), there is an infinite family of projectively unique \(d\)-dimensional polytopes \(\mathrm{PCCTP}^F_d[n]\subset S^d_+,\ n\ge 1,\) on \(12(n+1)+d+D(F)-9\) vertices with coordinates in \(F\), but not realizable (in \(S^d_+\)) with coordinates in \(G\), where \(G\) is any strict subfield of \(F\).

A point configuration or polytopal complex is rational if the coordinates of all of its points (resp. its vertices) are rational numbers.

Corollary 6.5.2

There is a \(D=D(\mathbb {Q})\) such that for any \(d \ge D\), there is an infinite family of rational projectively unique \(d\)-dimensional polytopes \(\mathrm{PCCTP}^{\mathbb {Q}}_d[n],\ n\ge 1,\) on \(12(n+1)+d+D-9\) vertices.

For the proof, let us first recall a fundamental result going back to von Staudt [47].

Proposition 6.5.3

(cf. [3, Cor. U.17]) Let \(Q\) denote any point configuration in \(S^d_+\subset S^d\) whose elements are described by algebraic coordinates. Then there is a projectively unique point configuration \(\mathrm{COOR}[Q]\) in \(S^d\) that contains \(Q\).

Corollary 6.5.4

For \(F\subset {\mathbb {R}}\) any finite field extension over \(\mathbb {Q}\), and for any rational point configuration \(Q\) in \(S^d_+\subset S^d\), \(d\ge 3\), there exists a projectively unique point configuration \(\mathrm{COOR}^F[Q]\in S^d_+\) that contains \(Q\) and such that \(\mathrm{COOR}^F[Q]\) is realizable with coordinates in \(F\), but not realizable with coordinates in \(G\), where \(G\) is any strict subfield of \(F\).

Proof of Theorem 6.5.1

As for Theorem II, the proof consists of two parts: We first give a family of rational polytopes \(\mathrm{CCTP}^{\mathbb {Q}}_4[n]\) for which \(\dim {\mathcal {R}\mathcal {S}}(\cdot )\) is uniformly bounded, and then produce projectively unique polytopes from them. \(\square \)

Construction of the polytopes \(\mathrm{CCTP}^{\mathbb {Q}}_4[n]\): We start by constructing CCTs analogous to those given in Sect. 4.1. Mirroring the notation of that section, let us replace \(\vartheta _0\) and \(\vartheta _1\) by

$$\begin{aligned} \vartheta ^\mathbb {Q}_0:=\left( \tfrac{1}{3}, -\tfrac{1}{3}, 2, 0, 1\right) \quad \text {and} \quad \vartheta ^\mathbb {Q}_1:=\left( 1, 0, \tfrac{3}{5}, 0, 1\right) , \end{aligned}$$

respectively. With this, we obtain \(\mathrm{T}^\mathbb {Q}\)[1], the starting CCT for our construction. We note, without proof, some facts about this complex:

  • The complex \(\mathrm{T}^\mathbb {Q}[1]\) can be extended twice to \(\mathrm{T}^\mathbb {Q}[3]\), which is ideal. Hence, Theorem 4.0.5 shows that \(\mathrm{T}^\mathbb {Q}[1]\) can be extended to \(\mathrm{T}^\mathbb {Q}[n]\) for any \(n\ge 1\).

  • The complex \(\mathrm{T}^\mathbb {Q}[3]\) is in convex position. Hence, the complexes \(\mathrm{T}^\mathbb {Q}[n],\ n\ge 1,\) are in convex position by Theorem 4.0.7.

  • Due to symmetry of ideal CCTs, \(\mathrm{T}^\mathbb {Q}[1]\) cannot be rational. However, it is linearly equivalent to a complex \(\varTheta (\mathrm{T}^\mathbb {Q}[1])\) with rational vertex coordinates. For instance, \(\varTheta \) can be chosen as the linear transformation given by the diagonal matrix with entries \((1,1,1,\sqrt{3},1)\).

  • As mentioned in Sect. A.4, the extensions of CCTs are given by a rational functions. Moreover, extension clearly commutes with any projective transformation. Any extension of the rational complex \(\varTheta (\mathrm{T}^\mathbb {Q}[1])\) is therefore rational.

Similar to Example 5.2.2, we give the coordinates of \(\mathrm{T}^\mathbb {Q}[n]\) up to layer \(10\) in Table 3 below. Again, we set \(\kappa _0:=\vartheta ^\mathbb {Q}_0\), \(\kappa _1:={\mathrm{r}}^2_{1,2}\vartheta ^\mathbb {Q}_1\), and use Proposition 6.4.3 and Formula 6.4.4 to inductively generate the complexes \(\mathrm{T}^\mathbb {Q}[n]\). We omit the fourth and fifth coordinate, since these are always \(0\) and 1, respectively.

Table 3 Coordinates for the polytopes \(\mathrm{CCTP}^{\mathbb {Q}}_4[n]\)

As in the case of the original cross-bedding cubical torus polytopes, we define the desired polytopes \(\mathrm{CCTP}^{\mathbb {Q}}_4[n]\) as convex hulls of the CCTs constructed:

$$\begin{aligned} \mathrm{CCTP}^{\mathbb {Q}}_4[n]:=\mathrm{conv}(\varTheta (\mathrm{T}^{\mathbb {Q}}_4[n])). \end{aligned}$$

With this, we have

$$\begin{aligned} f_0(\mathrm{CCTP}^{\mathbb {Q}}_4[n])=f_0(\mathrm{T}^{\mathbb {Q}}_4[n])=12(n+1), \end{aligned}$$

and hence, as in the proof of Theorem I,

$$\begin{aligned} \dim {\mathcal {R}\mathcal {S}}(\mathrm{CCTP}^{\mathbb {Q}}_4[n])\le 4f_0(\mathrm{T}^{\mathbb {Q}}[1])= 96. \end{aligned}$$

Construction of the polytopes \(\mathrm{PCCTP}^{F}_4[n]\): Set \(Q:=\varTheta (\mathrm{F}_0(\mathrm{T}^{\mathbb {Q}}[1]))=\mathrm{F}_0(\mathrm{CCTP}^{\mathbb {Q}}_4[1])\). Let \(V\) be a set of points in \(S^4_+\) that span a hyperplane not intersecting any of the polytopes \(\mathrm{CCTP}^{\mathbb {Q}}_4[n]\), similar to part VII. in the proof of Lemma 5.2.6. Apply Corollary 6.5.4 to find a projectively unique point configuration \(\mathrm{COOR}^F[Q\cup V]\) that contains \(Q\) and \(V\). By Lemma 4.4.2, \(\varTheta (\mathrm{F}_0(\mathrm{T}^{\mathbb {Q}}[1]))=\mathrm{F}_0(\mathrm{CCTP}^{\mathbb {Q}}_4[1])\) frames \(\mathrm{CCTP}^{\mathbb {Q}}_4[n]\) for all \(n\ge 1\). Hence

$$\begin{aligned} (\mathrm{CCTP}^{\mathbb {Q}}_4[n],Q,R),\ \ n\ge 1,\ \ R:=\mathrm{COOR}^F[Q\cup V]\setminus Q, \end{aligned}$$

is a weak projective triple. Realizability in vector spaces over fields is not affected by the operations subdirect cone (Definition 5.2.4 and Lemma 5.2.5) and Lawrence extension (Proposition 5.1.2), hence applying both yields the desired polytopes \(\mathrm{PCCTP}^{F}_{D(F)}[n]\). These polytopes have dimension \({D(F)}=f_0(Q)+f_0(R)+5=f_0(R)+29\) and the number of vertices computes to

$$\begin{aligned} f_0(\mathrm{PCCTP}^{F}_{D(F)}[n])&= 2(f_0(Q)+f_0(R))+f_0\left( \mathrm{CCTP}^{F}_4[n]\right) +1\\&= 12(n+1)+2D(F)-9. \end{aligned}$$

1.5.2 A.5.2 Many inscribed projectively unique polytopes

It is a classical and elementary fact that if \(W\) is a 3-cube and \(S\) is a sphere in \({\mathbb {R}}^d\) such that seven vertices of \(W\) lie in \(S\), then all vertices of \(W\) lie in \(S\) [26, Rec. 2]. In fact, if \(W\in {\mathbb {R}}^d\) is a 3-cube and \(\mathcal {Q}\) is a quadric that contains seven vertices of \({W}\), then the last vertex of \(W\) lies on \(\mathcal {Q}\) as well, cf. [7, Sec. 3.2]. As a consequence, we obtain the following beautiful result:

Proposition 6.5.5

Let \(\mathrm{T}[2]\) denote a CCT in \(S^d\) or \({\mathbb {R}}^d\), and let \(S\) be a sphere in \(S^d\) resp. \({\mathbb {R}}^d\) that contains \(\mathrm{F}_0(\mathrm{T}[2])\). Then for all extensions \(\mathrm{T}[n]\) of \(\mathrm{T}[2]\), we have \(\mathrm{F}_0(\mathrm{T}[n])\subset S\).

Here, a sphere in \(S^d\) is the intersection of \(S^d\) with some affine subspace of \({\mathbb {R}}^{d+1}\). This opens the door to the use of cross-bedding cubical tori in the theory of inscribed polytopes. A polytope in \(S^d\) or \({\mathbb {R}}^d\) is inscribed if all its vertices are contained in some sphere. Combinatorial types of polytopes that can be realized in an inscribed way are inscribable.

Inscribable polytopes are a classical and intriguing subject in polytope theory. Perhaps overly optimistic, Steiner [39] asked in 1832 for a classification of inscribable polytopes. For a long time, it was not even known whether all combinatorial types of polytopes are inscribable, until Steinitz provided an example of a non-inscribable polytope [41], cf. [16, Sec. 13.5]. Much later, interest in inscribed 3-polytopes experienced a revival due to their importance in the theory of hyperbolic 3-manifolds [44] and Delaunay triangulations [8]. Conversely, the connection to hyperbolic geometry led to an almost complete understanding of inscribable polytopes of dimension 3 [31, 32]. Many problems concerning inscribed polytopes remain; for some recent progress, compare [2, 14].

In this section, we present some progress in the direction of understanding high-dimensional inscribable polytopes by proving the following analogues of Theorems I and II for inscribed polytopes.

Theorem 6.5.6

For each \(d\ge 4\), there exists an infinite family of combinatorially distinct inscribed \(d\)-dimensional polytopes \(\mathrm{CCTP}^{\mathrm{in}}_d[n]\) with \(12(n+1)+d-4\) vertices such that \(\dim {\mathcal {R}\mathcal {S}}(\mathrm{CCTP}^{\mathrm{in}}_d[n])\le 76+d(d+1)\) for all \(n\ge 1\).

Theorem 6.5.7

There is a \(D\ge 0\) such that for each \(d\ge D\), there exists an infinite family of combinatorially distinct inscribed \(d\)-polytopes \(\mathrm{PCCTP}^{\mathrm{in}}_{d} [n],\, n\ge 1,\) with \(12(n+1)+d+D-9\) vertices, all of which are projectively unique.

By polar duality, analogous results hold for circumscribed polytopes (i.e. polytopes all whose facets are tangent to some sphere).

1.5.3 A.5.3 Many inscribed 4-polytopes with small realization space

Proof of Theorem 6.5.6

By Proposition 6.5.5, it suffices to provide an ideal CCT \(\mathrm{T}^{\mathrm{in}}[3]\) in convex position in \(S^4\) whose vertices lie in some sphere in \(S^4\). We do this analogously to Sect. 4.1. Parallel to the notation of that section, let us replace \(\vartheta _0\) and \(\vartheta _1\) by

$$\begin{aligned} \vartheta ^{\mathrm{in}}_0:=(1, -1, y, 0, 1)\quad \text {and} \quad \vartheta ^{\mathrm{in}}_1:=\big (\frac{11}{10}, x, z, 0, 1\big ), \end{aligned}$$

respectively. Here, \(x\) is defined as

$$\begin{aligned} x&= \frac{{{\left( 2 i \sqrt{3} \!-\! 2\right) } {\left( \!-\!45 i \sqrt{566805} \!+\! 83895\right) }^{\frac{2}{3}} 60^{\frac{2}{3}} \!-\! 263(i\sqrt{3}\!+\!1)60^{\frac{4}{3}} \!-\! 1560 {\left( -45 i \sqrt{566805} \!+\! 83895\right) }^{\frac{1}{3}}} }{3600 {\left( \!-\!45 i \sqrt{566805} \!+\! 83895\right) }^{\frac{1}{3}}} \\&= \frac{2^{\frac{5}{6}}}{60} \Big (2^{\frac{2}{3}} \sqrt{789} \sin \Big (\frac{1}{3} \arctan \Big (\frac{3\sqrt{566805}}{5593} \Big )\Big ) 2^{\frac{2}{3}} \sqrt{263} \cos \Big (\frac{1}{3} \arctan \Big (\frac{3\sqrt{566805}}{5593} \Big )\Big ) \!-\! 13 2^{\frac{1}{6}}\Big ) \end{aligned}$$

and we obtain \(y\) as

$$\begin{aligned} y = \frac{ 24 -20 x}{\sqrt{559-400x- 100 x^{2}}} \end{aligned}$$

and analogously

$$\begin{aligned} z=\frac{319-100 x^{2} - 200 x}{10\sqrt{559-400x- 100 x^{2}}} =\frac{{\left( 100 x^{2} + 200 x - 319\right) } y}{40(5x-6)}. \end{aligned}$$

This gives \(\mathrm{T}^{\mathrm{in}}[1]\), and \(\mathrm{T}^{\mathrm{in}}[3]\) is obtained by extending on \(\mathrm{T}^{\mathrm{in}}[1]\) twice. The coordinates of \(\vartheta ^{\mathrm{in}}_2\) and \(\vartheta ^{\mathrm{in}}_3\) are too complicated to give them here directly; indeed, it is not even trivial to see that the coordinates for \(\vartheta ^{\mathrm{in}}_0\) and \(\vartheta ^{\mathrm{in}}_1\), as given above, are real numbers. Their approximate values are given as

$$\begin{aligned} x\approx 1.0226363,\qquad y\approx 0.5266533,\quad \text {and}\quad z\approx 0.1468968. \end{aligned}$$

The CCT \(\mathrm{T}^{\mathrm{in}}[3]\) is an ideal CCT in convex position; in homogeneous coordinates, its vertices lie on a sphere with radius \(\approx 1.8103\). Theorems 4.0.5 and 4.0.7 demonstrate that \(\mathrm{T}^{\mathrm{in}}[3]\) can be extended to ideal CCTs \(\mathrm{T}^{\mathrm{in}}[n]\) in convex position in \(S^4\) of arbitrary width. Analogous to Example 5.2.2 and Section A.5.1, we give their coordinates in Table 4, this time only approximate and up to layer five, when exact calculations became infeasible. Again, we set \(\kappa _0:=\vartheta ^{\mathrm{in}}_0\), \(\kappa _1:={\mathrm{r}}^2_{1,2}\vartheta ^{\mathrm{in}}_1\), and inductively construct the complexes \(\mathrm{T}^{\mathrm{in}}[n]\). As before, we leave out the fourth and fifth coordinates.

Table 4 Coordinates for the polytopes \(\mathrm{CCTP}^{\mathrm{in}}_4[n]\)

As usual, we define

$$\begin{aligned} \mathrm{CCTP}^{\mathrm{in}}_4[n]:=\mathrm{conv}\, \mathrm{T}^{\mathrm{in}}[n]. \end{aligned}$$

These polytopes are inscribed for all \(n\ge 1\) because the vertices of \(\mathrm{T}^{\mathrm{in}}[n]\) lie in a common sphere by Proposition 6.5.5. Since

$$\begin{aligned} f_0(\mathrm{T}^{\mathrm{in}}[n])=f_0(\mathrm{CCTP}^{\mathrm{in}}_4[n])=12(n+1), \end{aligned}$$

Lemma 4.4.2 permits us to conclude the desired bound

$$\begin{aligned} \dim {\mathcal {R}\mathcal {S}}(\mathrm{CCTP}^{\mathrm{in}}_4[n])\le {\mathcal {R}\mathcal {S}}(\mathrm{T}^{\mathrm{in}}[1]) \le \dim {\mathcal {R}\mathcal {S}}(\mathrm{T}^{\mathrm{in}}[1])\le 4f_0(\mathrm{T}^{\mathrm{in}}[1])= 96. \end{aligned}$$

\(\square \)

Many inscribed projectively unique polytopes: What does remain is to construct inscribed projectively unique polytopes from the polytopes \(\mathrm{CCTP}^{\mathrm{in}}[n]\). The two key tools are the following observations on Lawrence extensions and subdirect cones, respectively. For an inscribed polytope \(P\), let us denote by \(\mathcal {U}(P)\) the circumscribed sphere to \(P\), and let \(\mathcal {B}(P):=\mathrm{conv}\, \mathcal {U}(P)\) denote the ball enclosed by it.

Proposition 6.5.8

(Lawrence extensions with regard to inscribed polytopes) Let \((P,R)\) denote a PP configuration in \(S^d\) such that \(P\) is an inscribed polytope and such that \(\mathcal {B}(P)\) contains no point of \(R\). Then the Lawrence polytope \(\mathrm{L}(P,R)\) of \((P,R)\) is inscribable.

We call such a PP configuration an inscribed PP configuration; a PP configuration is inscribable if it is Lawrence equivalent to some inscribed PP configuration.

Remark 6.5.9

The converse to Proposition 6.5.8 holds as well: If \((P,R)\) is not inscribable, then \(\mathrm{L}(P,R)\) is not inscribable either.

Proof

As in the case of Lawrence extensions (Proposition 5.1.2), we only treat the case where \(R=\{r\}\) consists of a single point. Recall that the Lawrence polytope \(\mathrm{L}(P,R)\) is obtained as

$$\begin{aligned} \mathrm{L}(P,R)=\mathrm{conv}P \cup \{\underline{r},\overline{r}\}, \end{aligned}$$

where \(\underline{r}\) and \(\overline{r}\) are points in \(S^{d+1}\supset S^d\) that lie in a common line with \(r\) such that \(\underline{r}\in [r,\overline{r}]\). If \(r\notin \mathcal {B}(P)\), then \(\underline{r}\) and \(\overline{r}\) can be chosen as the intersection points of some line containing \(r\) with \(\mathcal {S}\), where \(\mathcal {S}\) is any sphere in \(S^{d+1}\) with \(\mathcal {S}\cap S^d={\mathcal {U}(P)}.\) With this choice, all vertices of the Lawrence polytope \(\mathrm{L}(P,R)\) lie in \(\mathcal {S}\). \(\square \)

A similar result holds for weak projective triples and subdirect cones.

Lemma 6.5.10

(Subdirect cones with regard to inscribed polytopes) Let \((P,Q,R)\) denote any weak projective triple such that \(P\) is inscribed, and such that \(H\cap R\) does not intersect \(\mathcal {B}(P)\), where \(H\) denotes the wedge hyperplane of the weak projective triple. Then the subdirect cone \((P^v,Q\cup R)\) is an inscribable PP configuration. \(\square \)

Remark 6.5.11

If \((P,Q,R)\) is a weak projective triple such that no projective transformation of \((P,Q,R)\) satisfies the conditions of Lemma 6.5.10, then the subdirect cone \((P^v,Q\cup R)\) is not inscribable.

Remark 6.5.12

(Universality of inscribed polytopes and Delaunay triangulations) While simple observations, Proposition 6.5.8 and Lemma 6.5.10 can be applied to provide finer characterizations of realization spaces of inscribed polytopes. In [2], we apply them to derive the following results:

  • For every primary basic semialgebraic set \(S\) defined over \({\mathbb {Z}}\), there is an inscribed polytope (resp. Delaunay triangulation) whose realization space is homotopy equivalent to \(S\). This extends results of Mnëv [27] to the inscribed setting.

  • For every inscribed polytope \(P\), there is an inscribed, projectively unique polytope containing \(P\) as a face. In particular, there is a Delaunay triangulation that contains \(P\) and is unique up to similarities. This generalizes results of Padrol and the first named author [3].

Proof of Theorem 6.5.7

Set \(Q:=\mathrm{F}_0(\mathrm{CCTP}^{\mathrm{in}}_4[1])\). Then \(Q\) frames the polytopes \(\mathrm{CCTP}^{\mathrm{in}}_4[n]\) for any \(n\ge 1\) by Lemma 4.4.2, and its elements have algebraic coordinates by construction. Let \(V\) be any collection of points in \(S^4\) whose span does not intersect the ball \(\mathcal {B}(\mathrm{CCTP}^{\mathrm{in}}_4[1])=\mathcal {B}(\mathrm{CCTP}^{\mathrm{in}}_4[n])\) analogous to part VII. of the proof of Lemma 5.2.6. Consider the weak projective triples

$$\begin{aligned} \left( \mathrm{CCTP}^{\mathrm{in}}_4[n],Q,R)\right) ,\ \ n\ge 1,\ \ R:=\mathrm{COOR}[Q\cup V]\setminus Q, \end{aligned}$$

where \(\mathrm{COOR}[Q\cup V]\) is the projectively unique point configuration provided by Proposition 6.5.3, and let \(D=f_0(Q)+f_0(R)+5=f_0(R)+29\). The polytopes \(\mathrm{CCTP}^{\mathrm{in}}_4[n]\) are inscribed, hence, the subdirect cones of these triples are inscribable PP configurations (Lemma 6.5.10). The Lawrence extension of the PP configurations yield polytopes \(\mathrm{PCCTP}^{\mathrm{in}}_D[n]\), which are inscribable by Proposition 6.5.8 and projectively unique by Proposition 5.1.2. These polytopes have dimension \(D\) and satisfy

$$\begin{aligned} f_0\left( \mathrm{PCCTP}^{\mathrm{in}}_D[n]\right)&= 2(f_0(Q)+f_0(R))+f_0\left( \mathrm{CCTP}^{\mathrm{in}}_4[n]\right) +1\\&= 12(n+1)+2D-9. \end{aligned}$$

\(\square \)

1.6 A.6 Some technical details

1.6.1 A.6.1 Proof of Lemma 4.2.2

We now prove Lemma 4.2.2, which was used to prove that “locally” every ideal CCT admits an extension. We do so by first translating it into the language of dihedral angles, and then applying a local-to-global theorem for convexity. We stay in the notation of Lemma 4.2.2: \(\mathrm{T}\) is an ideal CCT of width at least 3 and \(\mathrm{T}^\circ :=\mathrm{R}(\mathrm{T},[k-2,k])\) denotes the subcomplex of \(\mathrm{T}\) induced by the vertices of the last three layers. Recall that \(\mathrm{T}^\circ \) is homeomorphic to \(S^1\times S^1\), and it is in particular a manifold that is not a sphere.

Definition 6.6.1

(Dihedral angle) Let \(M\) be a \(d\)-manifold with polytopal boundary in \({\mathbb {R}}^d\) or \(S^d\). Let \(\sigma ,\, \tau \) be two facets in \(\partial M\) that intersect in a \((d-2)\)-face. The (interior) dihedral angle at \(\sigma \cap \tau \) w.r.t. \(M\) is the angle between the hyperplanes spanned by \(\sigma \) and \(\tau \), respectively, measured in the interior of \(M\).

With this notion, we formulate a lemma that generalizes Lemma 4.2.2.

Lemma 6.6.2

Let \(M\) denote the closure of the component of \(S^3_{\mathrm{eq}}{\setminus } \mathrm{T}\) that contains \(\mathcal {C}_0\). Then the dihedral angles at edges in \(\mathrm{R}(\mathrm{T},[k-2,k-1])\subset \mathrm{T}^\circ \) w.r.t. \(M\) are strictly smaller than \(\pi \).

Lemma 6.6.2 implies Lemma 4.2.2

Let \(v\) denote any vertex of \(\mathrm{R}(\mathrm{T},k-2)\). By Lemma 6.6.2 all dihedral angles of \(\mathrm{Lk}(v,\mathrm{T}^\circ )\) are smaller than \(\pi \), so \(\mathrm{Lk}(v,\mathrm{T}^\circ )\) is the boundary of the convex triangle \({N}_v^1 M\subset {N}_v^1 S^3_{\mathrm{eq}}\), proving the first statement of Lemma 4.2.2. Furthermore, since \(\mathrm{T}\) is transversal, the tangent direction of \([v,\pi _0(v)]\) at \(v\) lies in \({N}_v^1 M=\mathrm{conv}\mathrm{Lk}(v,\mathrm{T}^\circ )\), proving the second statement. \(\square \)

The rest of this section is consequently dedicated to the proof of Lemma 6.6.2. We need the following elementary observation.

Observation 6.6.3

Consider the union \(s\) of three segments \([a,b]\), \([b,c]\) and \([a,c]\) on vertices \(a\), \(b\) and \(c\), and any component \(B\) of the complement of \(s\) in \(S^2\). Then the angles between the three segments w.r.t. \(B\) are either all smaller or equal to \(\pi \) or all greater or equal to \(\pi \). If \(a,\,b,\,c\) do not all lie on some common great circle, these inequalities are strict.

Proof of Lemma 6.6.2

Since the facets of \(\mathrm{T}\) are convex, all dihedral angles at vertices of layer \(k\) must be larger than \(\pi \). Thus the dihedral angles at vertices of layer \(k-2\) must be smaller or equal to \(\pi \), and at least one of the dihedral angles at each edge of layer \(k-2\) must be strictly smaller than \(\pi \), since the contrary assumption would imply that \(\mathrm{T}^\circ \) is the boundary of a convex body in \(S^3_{\mathrm{eq}}\) by Theorem 6.1.5, or the more elementary Theorem of Tietze [45, Satz 1] [28], and in particular homeomorphic to a 2-sphere, contradicting the assumption that \(\mathrm{T}^\circ \) is a torus.

Consider the star \(\mathrm{St}(v,\mathrm{T}^\circ )\) (Fig. 15) for any vertex \(v\) of \(\mathrm{R}(\mathrm{T},k-2)\). As observed, one of the dihedral angles at edges \([v,u]\), \([v,r]\) and \([v,q]\) must be smaller than \(\pi \). In particular, not all vertices of \(\mathrm{Lk}(v,\mathrm{T}^\circ )\) lie on a common great circle, and all dihedral angles at edges \([v,u]\), \([v,r]\) and \([v,q]\) w.r.t. \(M\) are smaller than \(\pi \) by Observation 6.6.3. \(\square \)

Fig. 15
figure 15

We show a part of the CCT \(\mathrm{T}^\circ \subset \mathrm{T}\). \(v\) is a vertex of layer \(k-2\), \(q,r,u\) are vertices of layer \(k-1\), and the vertices \(p,s,t\) lie in layer \(k\) of \(\mathrm{T}\)

1.6.2 A.6.2 Proof of Proposition 4.3.2

The goal of this section is to prove Proposition 4.3.2, which provides a local-to-global criterion for the convex position of ideal CCTs of width 3. We work in the 4-sphere \(S^4\subset {\mathbb {R}}^5\), and the equator sphere \(S^3_{\mathrm{eq}}\). Furthermore, \(\mathrm{p} \) shall denote the projection from \(S^4{\setminus } \{\pm e_5\}\) to \(S^3_{\mathrm{eq}}\).

Lemma 6.6.4

Let \(\mathrm{T}\) be an ideal 3-CCT in \(S^4\) such that for every facet \(\sigma \) of \(\mathrm{T}\), there exists a halfspace \(H(\sigma )\) containing \(\sigma \) in the boundary and such that \(H(\sigma )\) contains all remaining vertices of layers \(1,\, 2\) connected to \(\sigma \) via an edge of \(\mathrm{T}\) in the interior. Then \(H(\sigma )\) contains all vertices of \(\mathrm{F}_0(\mathrm{T}){\setminus } \mathrm{F}_0(\sigma )\) in the interior (Fig. 16).

Fig. 16
figure 16

The picture shows part of the complex \(\mathrm{R}(\mathrm{T},[0,2])\). Lemma 6.6.4 is concerned with the marked by black disks (connected to \(\sigma \) via an edge, in layers 1 and 2); Proposition 4.3.2 considers those vertices marked with an additional black circles (connected to \(\sigma \) via an edge and in layer 1)

Proof

We will prove the claim by contraposition. Let \(H=H(\sigma )\) denote a halfspace in \(S^4\) containing \(\sigma \) in the boundary, with outer normal \(\vec {n}=\vec {n}(H)\). To prove the claim of the lemma, assume that \(H^c:=S^4{\setminus } \mathrm{int} H\) contains a vertex \(w\) of \(\mathrm{T}\) that is not a vertex of \(\sigma \). We have to prove that there is another vertex that

  1. (i)

    lies in layers 1 or 2 of \(\mathrm{T}\),

  2. (ii)

    lies in \(H^c\),

  3. (iii)

    is not a vertex of \(\sigma \), but that

  4. (iv)

    is connected to \(\sigma \) via some edge of \(\mathrm{T}\).

We consider this problem in four cases. Let \(\sigma \) be any facet of \(\mathrm{T}\), and let us denote the orthogonal projection of \(x\in {\mathbb {R}}^5\) to the \(\mathrm{sp}\{e_i,\,e_j\}\)-plane in \({\mathbb {R}}^5\) by \(x_{i,j}\). By symmetry, \(\vec {n}=(*,\,*,\,\varepsilon v_3,\, \varepsilon v_4,\,*)\), where \(v=v(\sigma )=(v_1(\sigma ),\,\dots ,\,v_5(\sigma ))\) denotes the only layer \(0\) vertex of \(\sigma \), and \(\varepsilon =\varepsilon (H)\) is some real number. We can take care of the easiest case right away:

Case (0) If \(w\) lies in facet \({\tau }\) of \(\mathrm{T}\) that is obtained from \(\sigma \) by a rotation of the \(\mathrm{sp}\{e_3,\,e_4\}\)-plane, then \(\varepsilon =\varepsilon (H)\le 0\) by Proposition 3.3.1(b) and (d). Thus \(H^c\) must contain \({\tau }\) and all other facets of \(\mathrm{T}\) obtained from \(\sigma \) by rotation of the \(\mathrm{sp}\{e_3,\,e_4\}\)-plane. In particular, it contains all vertices of adjacent facets that are obtained from \(\sigma \) by a rotation of the \(\mathrm{sp}\{e_3,\,e_4\}\)-plane, among which we find the desired vertex, even a vertex satisfying (i) to (iv) among the vertices of \(\mathrm{R}(\mathrm{T},1)\). We may assume from now on that \(\varepsilon \) is positive.

It remains to consider the case in which \(w\) satisfies (ii) and (iii) and is obtained from a vertex of \(\sigma \) only from a nontrivial rotation of the \(\mathrm{sp}\{e_1,\,e_2\}\)-plane followed by a (possibly trivial) rotation of the \(\mathrm{sp}\{e_3,\,e_4\}\)-plane. Since \(\varepsilon \) is positive, there exists a vertex \(w'\) satisfying (ii) and (iii) in the same layer of \(\mathrm{T}\) as \(w\), but which lies in

$$\begin{aligned} \ell _0\!=\!\mathrm{p} ^{-1}\pi _2^{\mathrm{f}}(\mathrm{p} (v)),\ \ell _1\!=\!\mathrm{p} ^{-1}\pi _2^{\mathrm{f}}(\mathrm{p} (q))\!=\!{\mathrm{r}}_{3,4} \ell _0\ \text {or}\ \ell _{-1}\!=\!\mathrm{p} ^{-1}\pi _2^{\mathrm{f}}(\mathrm{p} (r))\!=\!{\mathrm{r}}_{3,4}^{-1}\ell _0. \end{aligned}$$

The existence of a vertex satisfying (i)-(iv) now follows from the following observation:

Let x,y be any two non-antipodal points in \(S^1\), and let m be any point in the segment [x,y]. Assume n is any further point in \(S^1\) such that \(\langle n,m\rangle \le \langle n,-m\rangle \). Then \(\langle n,y\rangle \le \langle n,-y\rangle \) or \(\langle n,x\rangle \le \langle n,-x\rangle \).

Fig. 17
figure 17

The three cases show how to, given a vertex satisfying (ii) and (iii) (circled black), obtain a vertex satisfying (i)–(iv) (found among the vertices marked by black disks)

Case (1) Assume \(w'\in \ell _0\). This case is only nontrivial if \(w'\) is not in \(\mathrm{R}(\mathrm{T},[1,2])\). Thus assume (w.l.o.g.) that \(w'\) lies in layer \(0\) (i.e. it is the vertex circled in Fig. 17(1)), the other case is fully analogous. Then \({\mathrm{r}}_{1,2}^2 w'=v\). To construct the desired vertex \(x\), note that \(\pi _2(\mathrm{p} (v))\) lies in the segment \([\pi _2(\mathrm{p} (u)),\pi _2(\mathrm{p} (p))]\) by Proposition 3.3.1(e). Now,

$$\begin{aligned} w'_{1,2}=-v_{1,2},\, w'_{3,4}=v_{3,4}\ \text {and}\ w'_{5}=v_{5}, \end{aligned}$$

and since \(w'\in H^c\), \(\langle \vec {n},v_{1,2}\rangle \le \langle \vec {n},w'_{1,2}\rangle \). Consequently, for \(x={u}\) or \(x={p}\), \(\langle \vec {n},x_{1,2}\rangle \le \langle \vec {n},-x_{1,2}\rangle \) and thus

$$\begin{aligned} \langle \vec {n},x\rangle \le \langle \vec {n},{\mathrm{r}}_{1,2}^2 x\rangle \ \Longleftrightarrow \ {\mathrm{r}}_{1,2}^2 x\in H^c. \end{aligned}$$

Since both \({\mathrm{r}}_{1,2}^2{u}\) and \({\mathrm{r}}_{1,2}^2{p}\) satisfy (i), (iii) and (iv), this vertex satisfies the properties (i) to (iv).

Case (2) Assume \(w'\in \mathrm{R}(\mathrm{T},[1,2])\cap (\ell _1\cup \ell _{-1})\). Then \(w'\) is obtained from a vertex \(y\) of \(\sigma \) by a rotation of the \(\mathrm{sp}\{e_1,\,e_2\}\)-plane, i.e. \(w'={\mathrm{r}}_{1,2}^2y\). It then follows, as in Case (1), that \({\mathrm{r}}_{1,2}^2{u}\) or \({\mathrm{r}}_{1,2}^2{p}\) lie in \(H^c\), since \(\pi _2(\mathrm{p} (y))\in [\pi _2(\mathrm{p} (u)),\pi _2(\mathrm{p} (p))]\) by Proposition 3.3.1(e). This vertex satisfies the properties (i) to (iv), because both \({\mathrm{r}}_{1,2}^2{u}\) and \({\mathrm{r}}_{1,2}^2{p}\) satisfy (i), (iii) and (iv).

Case (3) If \(w'\in \mathrm{R}(\mathrm{T},\{0\}\cup \{3\})\cap (\ell _1\cup \ell _{-1})\), then it must lie in a 2-face \(F\) which intersects \(\sigma \) in an 1-face (cf. Fig. 17 (3) for the case of layer \(0\) vertices). The remaining vertex of \(F\) that does not lie in \(\sigma \) is a vertex of layers 1 or 2. Since \(e\subseteq \partial H^c\) and \(w'\in H^c\), this vertex lies in \(H^c\) and must be connected to \(\sigma \) via an edge, and consequently satisfies properties (i) to (iv), as desired. \(\square \)

We can now prove Proposition 4.3.2.

Fig. 18
figure 18

(1) As before, the picture shows part of the complex \(\mathrm{R}(\mathrm{T},[0,2])\). We have to show that if all layer 1 vertices satisfying (iii) and (iv) lie in \(H(\sigma )\) (full black disks), then so do all vertices of layer 2 satisfying (iii) and (iv) (circled black). For the vertices of layer 2 marked by crosses, this follows as in Lemma 6.6.4. (2) The Illustration for inequality \(\mathrm{d}({p},q)\le \mathrm{d}(m',q)\)

Proof of Proposition 4.3.2

For the labeling of vertices, we refer to Fig. 18 (1). We stay in the notation of Lemma 6.6.4. Using its result, we only have to prove that if \(\sigma \) is a facet of \(\mathrm{T}\) and \(H(\sigma )\) with \(\sigma \subseteq \partial H(\sigma )\) is a halfspace such that the vertices \({\mathrm{r}}^2_{1,2}{u},\, {\mathrm{r}}^2_{3,4}{u}\) and \({\mathrm{r}}^{\,-2}_{3,4}{u}\) of \(\mathrm{R}(\mathrm{T},1)\subset \mathrm{T}\) lie in the interior of \(H(\sigma )\), then the vertices \({\mathrm{r}}^2_{1,2}{p},\) \( {\mathrm{r}}^2_{3,4}{p}\) and \({\mathrm{r}}^{\,-2}_{3,4}{p}\) of \(\mathrm{R}(\mathrm{T},2)\subset \mathrm{T}\) lie in \(\mathrm{int}H(\sigma )\) as well. Let \(\vec {n}=\vec {n}(\sigma )\) denote the outer normal to \(H(\sigma )\).

As already observed in the proof of Lemma 6.6.4, \(\vec {n}\) is of the form \(\vec {n}=(*,\, *,\,\varepsilon v_3,\, \varepsilon v_4,\, *)\), and if \({\mathrm{r}}^2_{3,4}{u}\in \mathrm{int}H(\sigma )\), then \(\varepsilon >0\) and

$$\begin{aligned} \langle \vec {n},{\mathrm{r}}^{\,\pm 2}_{3,4}{p}\rangle <\langle \vec {n},{p}\rangle \ \Longleftrightarrow \ {\mathrm{r}}^{\,\pm 2}_{3,4}{p}\in \mathrm{int}H(\sigma ) . \end{aligned}$$

Thus it remains to be proven that \(w={\mathrm{r}}^2_{1,2}{p}\) lies in \(\mathrm{int}H(\sigma )\), which we will do in two steps. If \(x\) and \(y\) lie in \(S^4{\setminus } \{z\in S^4:\ z_{1,2}=0\}\), let \(\mathrm{d}(x,y)\) measure the distance between \(\pi _0(\mathrm{p} (x))\) and \(\pi _0(\mathrm{p} (y))\) in \(S^4\).

  1. (a)

    We prove that \(\mathrm{d}({p},q)\) is smaller than \(\frac{\pi }{4}\).

  2. (b)

    From \(\mathrm{d}({p},q)<\frac{\pi }{4}\) we conclude that \(w={\mathrm{r}}_{1,2}^2 {p}\) lies in the interior of \(H(\sigma )\).

To see the inequality of (a), let \(m\) denote the midpoint of the segment \([r,q]\). Consider the unique point \(m'\) in \(\mathcal {C}_0\) such that \(\mathrm{p} (m)\in [\mathrm{p} ({u}),m']\), which is guaranteed to exist since \(\pi _2(\mathrm{p} ({u}))=\pi _2(\mathrm{p} (m))\) (Proposition 3.3.1(b) and (d)). We locate \(\mathrm{p} (p)\) w.r.t. \(\mathrm{p} ({m})\), \(\pi _0(\mathrm{p} ({m}))\) and \(m'\):

  • By Proposition 3.3.1(f), \(\mathrm{p} ({p})\) and \(m'\) lie in the same component of \(S^3_{\mathrm{eq}}{\setminus } \pi ^{\mathrm{sp}}_0(\mathrm{p} (q))\).

  • Furthermore, the complement of \(\mathrm{sp}\{\mathrm{p} (r),\, \mathrm{p} (q),\, \mathrm{p} ({u})\}\) in \(S^3_{\mathrm{eq}}\) contains \(\pi _0(\mathrm{p} (m))\) and \(\mathrm{p} ({p})\) in the same component, since \(\mathrm{p} (v)\) and \(\pi _0(\mathrm{p} (m))\) lie in different components.

  • Finally, by (b) and (d) of the same Proposition, \(\pi _2(\mathrm{p} (m))=\pi _2(\mathrm{p} ({p}))\).

Thus \(\mathrm{p} ({p})\) is contained in the triangle on vertices \(m'\), \(\pi _0(\mathrm{p} (m))\) and \(\mathrm{p} (m)\) in \(\pi _2^{\mathrm{sp}}(m)\), see also Fig. 18. As the projection of \(\mathrm{p} (m)\) to \(\mathcal {C}_0\) lies in the segment \([\pi _0(\mathrm{p} (m)),m']\), \(\pi _0(\mathrm{p} ({p}))\) lies in the segment from \(\pi _0(\mathrm{p} (q))=\pi _0(\mathrm{p} (m))\) to \(m'\).

Thus

$$\begin{aligned} \mathrm{d}({p},q)=\mathrm{d}({p},m)\le \mathrm{d}(m',m)=\mathrm{d}(m',q). \end{aligned}$$

To compute the latter, notice that, after applying rotations of planes \(\mathrm{sp}\{e_1,\,e_2\}\) and \(\mathrm{sp}\{e_3,\,e_4\}\), we may assume that

$$\begin{aligned} {u}=\big (u_1,0,u_3,0,1\big ),\ \ u_1,\, u_3>0. \end{aligned}$$

Then the coordinates of \(q\) and \(r\) are given as

$$\begin{aligned} \left( 0,-u_1,\tfrac{1}{2}u_3,\pm \tfrac{\sqrt{3}}{2}u_3,1\right) \ \quad \text {and} \quad m'=\tfrac{1}{\sqrt{5}}\big (1,-2,0,0,0\big ). \end{aligned}$$

In particular,

$$\begin{aligned} \mathrm{d}({p},q)\le \mathrm{d}(m',q)=\arctan \tfrac{1}{2}<\tfrac{\pi }{4}. \end{aligned}$$

As for step (b): After applying rotations of planes \(\mathrm{sp}\{e_1,\,e_2\}\) and \(\mathrm{sp}\{e_3,\,e_4\}\) of \(\mathrm{T}\), we may assume that \((u_1,0,u_3,0,1)\), as above. Then, as before, the coordinates of the remaining layer 1 vertices \(q\) and \(r\) of \(\sigma \) are \((0,-u_1,\frac{1}{2}u_3,\pm \frac{\sqrt{3}}{2}u_3,1)\). A straightforward calculation shows that any normal to \(\mathrm{conv}\{{u},\,q,\,r\}\) is a dilate of

$$\begin{aligned} \vec {n}=\big (\mu ,\,-(\mu +\tfrac{u_3}{2u_1}),\,1,\,0,\,n_5\big ),\ \ \mu \in {\mathbb {R}},\ n_5=-\mu u_1-u_3 \end{aligned}$$

and if \(\vec {n}\) is the outer normal to a halfspace \(H(\sigma )\) that exposes the triangle \(\mathrm{conv}\{{u},\,q,\,r\}\) among all vertices of \(\mathrm{R}(\mathrm{T},1)\) connected to \(\sigma \), then the dilation is by a positive real and \(\mu >0\).

Since \(\mathrm{T}\) is transversal, Proposition 3.3.1(e) gives \({p}_1<0\). If additionally \(\mathrm{d}({p},q)<\frac{\pi }{4}\), then \({p}_2<{p}_1<0\). In particular,

$$\begin{aligned} 2\mu ({p}_1-{p}_2)-\tfrac{u_3}{u_1}{p}_2>0, \end{aligned}$$

which, due to the fact that \(w_{1,2}=-{p}_{1,2},\, w_{3,4}={p}_{3,4}\) and \(w_{5}={p}_{5}=1\), is equivalent to

$$\begin{aligned} \langle \vec {n},{p}\rangle&= {p}_1 \mu -\big (\mu +\tfrac{u_3}{2u_1}\big ){p}_2+{p}_3+n_5>-{p}_1 \mu + \big (\mu +\tfrac{u_3}{2u_1}\big ){p}_2+{p}_3+n_5\\&= \langle \vec {n},w\rangle , \end{aligned}$$

and consequently \(w\) is in the interior of \(H(\sigma )\). \(\square \)

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Adiprasito, K.A., Ziegler, G.M. Many projectively unique polytopes. Invent. math. 199, 581–652 (2015). https://doi.org/10.1007/s00222-014-0519-y

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