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On isometric embeddings of prisms N. P. Dolbilin, M. I. Shtogrin Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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     § 1. IntroductionIn Pogorelov’s book [1] (Ch. 8, § 4 “Isometric transformations of a cylindrical surface satisfying the condition of support on circles at the edges”) several solutions were proposed for the problem formulated in the title of that section (see [1], Figs. 47 and 48 on pp. 213 and 214). However, it was shown in [4] that the embeddings of a cylindrical surface that are shown in Figs. 47 and 48 in [1] are not isometric. The geometric problem considered by Pogorelov can be formulated as follows: construct a nontrivial isometric piecewise smooth embedding of the surface of a right circular cylinder of finite height in three-dimensional space $\mathbb R^3$ that satisfy the following three conditions (Pogorelov’s conditions):
In [1], Ch. 8, § 2, Pogorelov constructed a family of nontrivial isometric embeddings of an infinite circular cylinder which are invariant under translations along the cylinder axis and rotations of finite order around it. Shtogrin [2] used this family to identify a subclass of isometric embeddings of a finite circular cylinder which satisfy the last of the three conditions, but do not satisfy the first two. More precisely, in [2] isometric embeddings of regular even-angled prisms were constructed that satisfy modified Pogorelov’s conditions, under which the bases of the cylinder are not circles, but regular polygons with an even number of edges. On the other hand, as follows from [3], there are isometric embeddings of a circular cylinder that satisfy the first two conditions, but not the third, that is, the embedding has an infinite number of (smoothness) components. At the same time it is not known whether there exists a nontrivial isometric piecewise smooth embedding of the lateral surface of a right circular cylinder that satisfies all three Pogorelov conditions. In this paper we construct nontrivial isometric embeddings of right prisms whose bases are arbitrary convex polygons, and not just regular polygons, as it was the case in [2]–[4]. Namely, given a right $n$-gonal prism, we propose a construction of isometric transformations that satisfy the following conditions (1)–(3), which we also refer to as Pogorelov’s conditions:
§ 2. Construction: the $\beta$-format of a polygon and the associated prismLet $P$ be a convex $n$-gon $A_1A_2\dots A_n$: see Figure 1, (a). For a sufficiently small positive number $\beta$, we construct a new polygon $P_\beta$ with $2n$ edges. To do this, we plot a point $a_i$ on the bisector of each angle $\angle A_i$ so that $|A_ia_i|=\beta$: see Figure 1, (b), for $i=1$. Then we draw a line $B_iD_i\perp A_ia_i$ through $a_i$. For a sufficiently small $\beta$, the line $B_iD_i$ intersects the edges $A_iA_{i+1}$ and $A_1A_{i-1}$ of $P$ adjacent to $A_i$. We assume that the points $B_i$ and $D_i$ on this line have the same distance $\beta$ to the adjacent edges $A_iA_{i+1}$ and $A_iA_{i-1}$, respectively: $|B_ib_i| =|D_id_i|=|A_ia_i|=\beta$; see Figure 1, (b). It is clear that $B_i$ and $D_i$ lie outside $P$. Now we connect the points $B_i$ and $D_{i+1}$, $i\in 1,\dots,n-1$, as well as $B_n$ and $D_1$, by line segments. We denote the resulting $2n$-gon $D_1B_1D_2B_2\dots D_nB_n$ by $P_\beta$ and refer to it as the $\beta$-format of the polygon $P$. Note that for sufficiently small $\beta$ the $\beta$-format $P_\beta$ is a non-self-intersecting convex polygon. Let us formulate more precisely the conditions under which the $\beta$-format is a convex $2n$-gon. Let $\angle A_i=\alpha_i$. Then it is easy to see that in the isosceles triangle $\triangle B_iA_iD_i$, The distance $|A_i b_i|$ of $A_i$ to the projection $b_i$ of $B_i$ onto the edge $A_iA_{i+1}$ is given by
$$
\begin{equation*}
|A_i b_i|=\beta \frac{1+\sin(\alpha_i/2)}{\cos(\alpha_i/2)} =\beta\tan\frac{\pi +\alpha_i}{4};
\end{equation*}
\notag
$$
see Figure 1. Note that the coefficient $\tan((\pi +\alpha_i)/4)$ is positive since $(\pi +\alpha_i)/4<\pi/2$. For sufficiently small $\beta$ the projection $b_i$ of $B_i$ onto $A_iA_{i+1}$ is closer to the vertex $A_i$ than the projection $d_{i+1}$ of $D_{i+ 1}$. Hence the points $A_i,b_i,d_{i+1},A_{i+1}$ are positioned on the edge $A_iA_{i+1}$ in just this order. Since for small $\beta$ this condition is satisfied for all edges $A_iA_{i+1}$ of the polygon $P$, the resulting $2n$-gon $D_1B_1D_2B_2\dots D_nB_n$ is convex. For sufficiently small $\beta$, as just noted, the points $b_i$ and $d_{i+1}$ are such that for each $i\in 1,\dots,n-1$ the length of the edge $B_iD_{i+1}$ of the $\beta$-format $P_\beta $ is equal to
$$
\begin{equation}
|B_iD_{i+1}|=|b_id_{i+1}|=|A_iA_{i+1}|-\beta \biggl(\tan\frac{\alpha_i+\pi}{4} +\tan\frac{\alpha_{i+1}+\pi}{4} \biggr)>0.
\end{equation}
\tag{1}
$$
For $i=n$ condition (1) can be written as
$$
\begin{equation}
|B_nD_1|=|b_n d_1 |=|A_nA_1| - \beta \biggl(\tan\frac{\alpha_n+\pi}{4} +\tan\frac{\alpha_1+\pi}{4} \biggr)>0.
\end{equation}
\tag{$1'$}
$$
This implies the following result. Lemma 2.1. Given a polygon $P$, for any $\beta>0$ satisfying inequalities (1) and ($1'$) there exists a convex $\beta$-format $P_\beta=D_1B_1D_2B_2\dots D_nB_n$ of $P$ with precisely $2n$ edges. For any $i\in 1,\dots,n$, we denote by $\beta_i$ the unique number for which inequality (1) (or ($1'$)) turns to equality. Let $\widehat \beta=\min_i \beta_i $. Then the statement of Lemma 2.1 can be refined as follows. Lemma 2.2. Given a convex $n$-gon $P$, for each $\beta\in (0, \widehat \beta)$ there is a $\beta$-format of $P$ with $2n$ edges. For $\beta=0$, the $\beta$-format degenerates into an $n$-gon $P$. When $\beta=\widehat \beta$, at least one of the $n$ edges of $P_\beta$ parallel to edges of $P$ collapses to a point, but $P_\beta $ remains a convex polygon. Proof. Take an edge $A_iA_{i+1}$ of $P$ and the corresponding polygonal line $a_iB_iD_{i+1}a_{i+1}$ lying on the boundary of $P_\beta$; see Figure 1, (c), for $n=4$. The right triangles $\triangle A_iB_i b_i$ and $\triangle B_iA_ia_i$ (Figure 1, (b), for $i=1$) have a common hypotenuse $A_iB_i$ and both of their legs $B_ib_i$ and $A_ia_i$ are of length $\beta$; see Figure 1. Therefore, the other two legs are also equal: $|a_iB_i|=|A_i b_i|$. For the same reason we have $|a_{i+1}D_{i+1}|=|A_{i+1} d_{i+1}|$. Hence $|A_iA_{i+1}|=|a_iB_i|+|B_iD_{i+1}|+|D_{i+1}a_{i+1}|$. This is true for each of the edges $A_1A_2,\dots,A_{n-1}A_n, A_n A_1$ of $P$. The lemma is proved. We say that $\beta>0$ is admissible if there exists a $2n$-gonal $\beta$-format $P_\beta$. For each vertex $A_i$ of $P_\beta$ consider the angle $\angle B_iA_i D_i$. The intersection of all these angles is a convex $2n$-gon; see Figure 2. The angle $\angle B_iA_i D_i$ does not depend on $\beta $:Thus, the $2n$-gon $Q$ is uniquely determined by the $n$-gon $P$. We refer to $Q$ as the associated polygon of $P$. Note that the $\beta$-format $P_\beta$ of the polygon $P$ is a closed billiard trajectory in the associated polygon $Q$. For ease of reference we state a lemma that follows immediately from the previous lemmas. Lemma 2.4. Let $P=A_1A_2\dots A_n$ be a convex $n$-gon, and let $P_\beta$ be a $\beta$-format of $P$. Let $a_iB_iD_{i+1}a_{i+1}$ be the fragment of $P_\beta$ corresponding to the edge $A_iA_{i+1}$, where the points $a_i$ and $ a_{i+1}$ lie on the bisectors of the angles $\angle A_i$ and $\angle A_{i+1}$, and $| A_ia_i|=|A_{i+1}a_{i+1}|=\beta$: see Figure 3. Choose a point $\widehat a_{i+1}$ on the straight line $a_iB_i$ such that $|a_i \widehat a_{i+1}|=|A_iA_{i+1}|$, and chose points $a_i'$ and $a''_{i+1}$ on the line $B_iD_ {i+1}$ such that $A_i a'_i a''_{i+1}A_{i+1}$ is a rectangle. Then the line segments $[a_i' a''_{i+1}]$ and $[a_i \widehat a_{i+1}]$ are symmetric with respect to the line $A_iC_i$, and the segments $[D_{i+1} a''_{i+1}]$ and $[D_{i+1} a_{i+1}]$ are symmetric with respect to $C_i A_{i+1}$. Proof. The $\beta$-format $P_\beta$ is a closed billiard trajectory $D_1B_1\dots D_nB_n$ in the $2n$-gon $Q=A_1C_1\dots A_nC_n$. Moreover, the segments $[B_iD_{i+1}]$ of this trajectory are parallel to the corresponding edges $A_iA_{i+1}$ of $P$ and lie at distance $\beta$ from them. The segments $[B_ia_i]$ and $[D_{i+1}a_{i+1}]$ are orthogonal to the bisectors of the angles $\angle A_i$ and $\angle A_{i+1}$, respectively, and $|a_iA_i|= |A_{i+1}a_{i+1}|=\beta$. Furthermore, the lines $A_iC_i$ and $C_iA_{i+1}$ are the bisectors of the angles $\angle a_iB_ia_i'$ and $\angle a_{i+1}D_{i+1} a''_{i+1}$; see Figure 3. This immediately implies the symmetry of the segments $[a_i' a''_{i+1}]$ and $[a_i \widehat a_{i+1}]$ with respect to $A_iC_i$.
 Figure 3.To Lemma 2.4: superposing the edge $A_iA_{i+1}$ on a fragment of the $\beta$-format $P_\beta$. To establish the symmetry of $[D_{i+1} a''_{i+1}]$ and $[D_{i+1} a_{i+1}]$ with respect to $C_i A_{i+1}$ we note that $\angle D_{i+1}a_{i+1}A_{i+1}=90^\circ=\angle D_ {i+1} a_{i+1}'' A_{i+1}$ by the construction of the $\beta$-format. Now the symmetry of the points $a''_{i+1}$ and $a_{i+1}$ with respect to $C_iA_{i+1}$ follows from the equality $\triangle D_{i+1} a_{i+1} A_{i+1}=\triangle D_{i+1} a''_{i+1}A_{i+1}$. The lemma is proved. We denote by $\Pi$ a right prism of height $H$ with base a convex polygon $P$. The prism $\Pi$ is a closed surface consisting of $n+2$ flat faces: $n$ lateral rectangular faces and two bases. We also consider the right prism $\Pi_Q$ of the same height $H$ whose base is the associated polygon $Q$. We say that the prism $\Pi_Q$ is associated with the prism $\Pi$. Note that any lateral face of the associated prism $\Pi_Q$ has two lateral edges on its boundary, one of which is the main edge, belonging to the prism $\Pi$. The constructions of isometric maps of a right prism $\Pi$ that follow use the associated prism. § 3. Main resultLet $\Pi$ be a right prism with base $P=A_1A_2\dots A_n$ and height $H$. We refer to a closed piecewise regular surface as a Pogorelov isoprism if it is the image of the prism $\Pi$ under a piecewise smooth isometric map $f$ satisfying the following conditions:
The distance $h$ between the planes of the polygons $f(P)$ and $f(P')$ is called the height of the isoprism $f(\Pi)$. Note that by condition (2) the convex hull $\operatorname{conv}(f( P)\cup f(P'))$ of the union $f(P)\cup f(P')$ is a right prism with bases $ f(P)$ and $f(P')$, and its height is equal to the height $h$ of the isoprism $f( \Pi)$. Theorem 3.1. Let $\Pi$ be a right prism with base $P$ and height $H$. Then for each $h$, $0<h\leqslant H$, there exists a Pogorelov isoprism $\widehat \Pi$ of height $h$. This isoprism is determined by a function $\beta (t)$ which is continuous and piecewise smooth on the interval $[0,h]$, $0\leqslant t\leqslant h\leqslant H$, and satisfies the following conditions: Proof. To avoid wordy (albeit very simple) arguments in the cases when $\beta$ reaches the maximum $\widehat \beta $ or is equal to zero, we consider only the case when $\beta(t) \leqslant \widehat \beta $ for all $t\in [0,h]$ and $\beta (t)=0$ only for $t=0$ and $t=h$.
For each $i\in 1,\dots,n$ consider the dihedral angle of the prism $\Pi$ at the lateral edge $A_iA_i'$ and introduce an orthogonal coordinate system $(t, \beta )$ on its bisector plane ${\mathcal B}_i$; see Figure 4, (a). In this coordinate system we consider the piecewise smooth graph of the function $\beta(t)$ satisfying conditions (1)–(3) above.  Figure 4.To Theorem 3.1: (a) the graph of the function $\beta(t)$ in the bisector plane ${\mathcal B}_i$; (b) the construction of the Pogorelov isoprism. Note that by condition (3) the equality $h=H$ is possible only for $\beta(t)\,{\equiv}\, 0$, when the isoprism $f(\Pi)$ is congruent to the prism $\Pi $. Let the rectangle $ A_iA_i'A_{i+1}'A_{i+1}$ be the $i$th lateral face of the prism $\Pi$. This face corresponds to a pair of lateral faces $A_iA_i'C_i'C_i$ and $C_iC_i'A_{i+1}'A_{i+1}$ of the associated prism $\Pi_Q$. We denote the planes containing these faces by ${\mathcal F}_i$ and ${\mathcal F}_{i+1}$, respectively. We assume that the curves $\beta(t)$ in the bisector planes of the dihedral angles at all lateral edges of the prism $\Pi$ are pairwise congruent. Note that the plane ${\mathcal B}_i$ is also the bisector plane for the dihedral angle of the associated prism $\Pi_Q$, and this angle $\angle C_{i-1}A_iC_i$ is equal to $(\pi+\alpha_i)/2$; see Figure 4, (b). We denote by $Z_i=Z_i(\beta)$ the cylindrical surface whose directrix is the plane curve $\beta(t)\subset\mathcal{B}_i$ and whose generatrix $L_i(t)$ is the segment $a_i(t) \widehat{a}_{i+1}(t)$, where $|a_i(t)\widehat{a}_{i+1}(t)|=|A_i(t)A_{i+1} (t)|=|A_iA_{i+1}|$ and $a_i(t)\widehat{a}_{i+1}(t)\bot\mathcal{B}_i$: see Figure 5. Since the length of the curve $\beta(t)$, $0\leqslant t\leqslant h$, is equal to $H$ (condition (3)), the cylindrical surface $Z_i$ is isometric to the lateral face $A_iA'_iA'_{i+ 1}A_{i+1}$ of the prism $\Pi$.  Figure 5.To the proof of Theorem 3.1: the transformation of the generatrix segment $a_i(t)\widehat a_{i+1}(t)$ into a polygonal line $a_i(t)B_i(t)D_{ i+1}(t)a_{i+1}(t)$ on the $\beta$-format $P_{\beta (t)}$. By virtue of conditions (1) and (2) in Theorem 3.1, the plane $\mathcal F_i$ of the associated prism $\Pi_Q$ cuts the generatrix $L_i(t)$, $t\in (0,h)$, into two line segments, and cuts the whole cylindrical surface $Z_i$ into two cylindrical components, the ‘interior’ one (with respect to the prism $\Pi_Q$) with generatrices $[a_i(t) B_i(t)]$ (we denote it by $Z^+_i$) and the ‘exterior’ one with generatrices $[B_i(t)\widehat a_{i+1}(t)]$. We reflect the exterior component in the plane ${\mathcal F}_i$. Its mirror image is a cylindrical surface with generatrices $[B_i(t) a''_{i+1}(t)]$. The directrix orthogonal to them is mirror symmetric to the curve $\beta (t)$. The plane $\mathcal F_{i+1}$ of the adjacent face of $\Pi_Q$ also intersects the generatrix $B_i(t)a''_{i+1}(t)$ at some point $D_{i+1}(t)$ and therefore cuts the mirror image of the cylinder into two components. We denote the first of them, with generatrix $B_i(t)D_{i+1}(t)$, by $Z^-_i$. We reflect the second cylindrical component with generatrix $[D_{i+1}(t)a''_{i+1}]$ in the plane $\mathcal F_{i+1}$ to obtain a cylindrical component $\widetilde Z_{ i+1}^+$ with generatrix $D_{i+1}(t)a_{i+1}(t)$. The surface consists of three cylindrical components, intersecting pairwise along the curves $\gamma_i(t)\subset \mathcal F_i$ and $\gamma_{i+1}(t)\subset \mathcal F_{i+1}$, where the functions defining these curves are expressed in terms of $\beta (t)$ as follows:
$$
\begin{equation}
\gamma_i(t)=\frac{\beta (t)}{\cos((\alpha_i+\pi)/4)}\quad\text{and} \quad \gamma_{i+1}(t)=\frac{\beta (t)}{\cos((\alpha_{i+1}+\pi)/4)}.
\end{equation}
\tag{3}
$$
The function $\gamma_i(t)$ in (3) depends on the angle $\alpha_i$. The surface $\Phi_i$ is isometric to the cylindrical surface $Z_i$, and therefore to the $i$th lateral face $A_iA_i'A_{i+1}'A_{i+1}$ of the prism $\Pi$. Note that the surfaces
$$
\begin{equation*}
\Phi_{i-1}=Z^+_{i-1}\cup Z^-_{i-1}\cup \widetilde Z^+_i\quad\text{and} \quad \Phi_i=Z_i^+\cup Z_i^- \cup \widetilde Z^+_{i+1}
\end{equation*}
\notag
$$
intersect along a common curve $\beta $ lying in the bisector plane $\mathcal B_i$:
$$
\begin{equation*}
\Phi_{i-1}\cap \Phi_i=\widetilde Z_i^+ \cap Z_i^+=\beta \subset \mathcal B_i.
\end{equation*}
\notag
$$
Identifying $\Phi_{i-1}$ and $\Phi_i$ along the common boundary $\beta (t) \subset \mathcal B_i$ for each $i\in 1,\dots,n$, we obtain a surface $\Phi $ isometric to the lateral surface of the prism $\Pi$, where The theorem is proved.Remark 3.1. Condition (1) in Theorem 3.1 on the behaviour of the function $\beta (t)$ at the interior points of the interval $(0,h)$ can easily be weakened to $0\leqslant \beta (t)\leqslant \widehat \beta $. Note that if $\beta (t_0)=0$ for $0<t_0<h$, then the plane section $t=t_0$ of the isoprism $f(\Pi)$ is an $n$-gon equal and parallel to $P$. If $\beta (t_0)=\widehat \beta$, then the corresponding plane section is an $m$-gon, where $n \leqslant m < 2n $. Remark 3.2. The resulting surface $\Phi$ is a tubular surface in the sense of Pogorelov, just like the isometric embeddings of a circular cylinder presented in [1], Ch. 8, § 2. § 4. A generalization: $\beta$-prismatoidsTheorem 3.1 can be extended to a wider class of polyhedra, which we refer to as $\beta$-prismatoids. Recall that a convex polyhedron is called a prismatoid if its vertices lie in two parallel planes. The bases of a prismatoid are the polygons obtained as the convex hulls of these two groups of vertices. Lateral faces of the prismatoid are triangles or trapezoids. The distance $H$ between the planes of its bases is the height of the prismatoid. We describe a special class of prismatoids. Consider a right prism $\Pi$ with base a convex polygon $P=A_1A_2\dots A_n $. Then construct a $\beta$-format $P_{\beta_1}$ of the lower base $P$ with parameter $\beta=\beta_1$ and a $\beta$-format $P'_{\beta_2}$ of the upper base $P'=A_1'A_2'\dots A_n'$ with parameter $\beta=\beta_2$. The convex hull of the union $P_{\beta_1}\cup P_{\beta_2}'$ is a prismatoid, and all of its $2n$ lateral faces are trapezoids if $0<\beta_j<\widehat \beta$, $j=1,2$. Moreover, the trapezoids corresponding to the lateral edges $A_iA_i'$ of the prism $\Pi$ are equilateral. In the case when at least one of the numbers $\beta_j$, $j=1,2$, is equal to $0$ or $\widehat \beta$ some trapezoids degenerate into triangles or segments. The boundary of the polyhedron $\operatorname{conv}(P_{\beta_1}\cup P'_{\beta_2})$ is referred to as a $\beta$-prismatoid and is denoted by $\Pi_{\beta_1\beta_2}$; see Figure 6. Next we describe a class of isometric Pogorelov-type transformations of $\beta$-prismatoids. We choose an arbitrary vertex $A_i$ in the original convex polygon $P$. It corresponds to a point $a_i$ of the $\beta$-format $P_{\beta_1}$: see Figure 6. Similarly, the vertex $A_i'$ corresponds to a point $a_i'$ on the upper base $P_{\beta_2}'$ of the prismatoid. The line segment $[a_i,a_i']$ is the median of a trapezoid, the median of a triangle, or an edge if none of the values $\beta_1$ and $\beta_2$ vanishes, or just one of them vanishes, or both vanish, respectively. We say that an isometric map $f$ is a Pogorelov-type isometry of the $\beta$-prismatoid $\Pi_{\beta_1\beta_2}$ if the following conditions are satisfied:
We consider a continuous piecewise smooth nonnegative function $\beta (t)$ on the interval $[0,h]$, $0< h\leqslant H, $ which satisfies the conditions
Theorem 4.1. Let $\Pi$ be a right prism with base an $n$-gon $P$, and let $\Pi_{\beta_1\beta_2}$ be a $\beta$-prismatoid with bases $P_{\beta_1}$ and $P'_{\beta_2}$; see Figure 6 (a). Let $\beta(t)$ be a curve (the graph of a nonnegative function $\beta(t)$ satisfying the conditions (a) and (b)) in the bisector plane $\mathcal B_i$ of the dihedral angle at the lateral edge $A_iA_i'$ of $\Pi$. Then there exists a Pogorelov-type isometry $f$ of the $\beta$-prismatoid $\Pi_{\beta_1\beta_2}$, which is uniquely determined by the curve $\beta (t)$; see Figure 6, (b). Clearly, Theorem 4.1 can be proved in the same way as Theorem 3.1. § 5. ConclusionHere we show that by iterating the construction of the $\beta$-format we can significantly expand the set of isometric transformations of a right $n$-gonal prism. All these isometries satisfy Pogorelov’s condition of ‘support on congruent $n$-gons at the edges’. Let $f(\Pi )$ be the Pogorelov isoprism which is constructed from a fixed right prism $\Pi$ and a function $\beta(t)$ satisfying the conditions of Theorem 3.1. Note that these conditions imply that for each $t_1\in (0,h)$ such that $0<\beta(t_1)<\max_{0\leqslant t\leqslant h} \beta (t)$ there exists $t_2$, $t_2\neq t_1$ (we can assume that $t_1<t_2$) such that $\beta(t_1)=\beta(t_2):=\beta_0$, where $\beta_0 \neq $0. Consider the plane $t_1$- and $t_2$-cross-sections of the Pogorelov isoprism $f(\Pi)$. Both the sections $P_{\beta_0}$ and $P'_{\beta_0}$ are equal $2n$-gons, which are parallel to each other, namely, they are the $\beta$-formats of the polygon $P$ for $\beta= \beta_0$. Moreover, each of these $2n$-gons projects orthogonally onto the other. These two cross-sections partition the lateral surface of the isoprism $f(\Pi)$ into three components $S_1$, $\widetilde S$ and $S_2$, where $S_1$ corresponds to the interval $[0,t_1]$, $\widetilde S$ corresponds to $[t_1,t_2]$ and $S_2$ to $[t_2,h]$. The components $S_1$, $\widetilde S$ and $S_2$ are isometric to the components $\mathcal S_1$, $\widetilde {\mathcal S}$ and $\mathcal S_2$ of the lateral surface of $\Pi$, with height between $0$ and $h_1$, $h_1$ and $h_2$, and $h_2$ and $H$, respectively; see Figure 7. The heights of these components, $h_1$, $h_2-h_1$ and $H-h_2$, are equal to the lengths of the curve $\beta(t)$ on the intervals $[0,t_1]$, $[ t_1,t_2]$ and $[t_2,h]$, respectively. The component $\widetilde S$ of the isoprism $f(\Pi )$ has two boundaries, $P_{ \beta_0}$ and $P_{ \beta_0}'$, which are $\beta$-formats for $\beta=\beta_0$ . The component $\widetilde S$ is isometric to the component $\widetilde{\mathcal S}$ of the lateral surface of the prism $\Pi$. The component $\widetilde{\mathcal S}$ with $n$-gonal base $P$ is isometric to the lateral surface (denoted by $\widetilde T$) of the right prism $\Pi_{2n}$, whose base is the $\beta$-format $P_{\beta_0}$ (a $2n$-gon) and whose height is $h_2-h_1$. Indeed, both surfaces $\widetilde{\mathcal S}$ and $\widetilde T$ are the lateral surfaces of right prisms of the same height, and their bases are the isoperimetric polygons $P$ and $P_{\beta_0}$. Consider a continuous piecewise smooth nonnegative function $\widetilde \beta (t)$, $t\in [t_1, t_2] $, such that the length of the curve $\widetilde\beta(t)$ is equal to the height $h_2-h_1$ of the right $2n$-gonal prism $\Pi_{2n}$ with lateral surface $\widetilde T$; see Figure 7, (c). The length of $\widetilde{\beta}(t)$ is equal to the length of the curve $\beta(t)$ on the middle interval $[t_1, t_2]$. By Theorem 3.1 the function $\widetilde\beta (t)$, $t\in [t_1,t_2]$, determines an isometric transformation $f'$ of the lateral surface $\widetilde T$ of $\Pi_{2n}$. The surface $\mathcal U$ (see Figure 7, (d)) is isometric to the lateral surface $\widetilde T$ of $\Pi_{2n}$ (see Figure 7, (c)). The surface $\widetilde T$ is isometric to the middle part $\widetilde{\mathcal S}$ of the lateral surface of $\Pi$ (see Figure 7, (a)). Since $\widetilde S= f(\widetilde{\mathcal S})$, the surface $\mathcal U$, which is isometric to the component $\widetilde{\mathcal S}\subset \Pi $, is also isometric to the component $ \widetilde S \subset f(\Pi)$. Moreover, the boundaries of the components $\widetilde S$ and $\mathcal U$ are congruent $2n$-gons ($\beta$-formats of $P$ for $\beta= \beta_0$) lying at the constant distance $t_2-t_1$ from each other. Therefore, the component $\widetilde S$ of the lateral surface of the isoprism $f(\Pi)$ can be replaced by the lateral surface $\mathcal U$ of the isoprism $f'(\widetilde T)$ with natural identification of the corresponding boundaries: see Figure 7, (e). As a result, we obtain a closed surface $g(\Pi)$ isometric to the surface of the prism $\Pi$. The isometry $g$ of a right $n$-gonal prism $\Pi$ differs from the isometries $f$ described in Theorem 3.1. In the latter case, any ‘horizontal’ section of the isoprism $f( \Pi )$ is a $\beta$-format of the $n$-gon $P$, that is, an $2n$-gon (for $\beta>0$). Whereas in the isoprism $g(\Pi)$ there is a family of sections that are $\beta$-formats of the $2n$-gon $P_{\beta_0}$, which are $4n$-gons. Thus, the isoprism $g(\Pi)$ has both $2n$-gonal and $4n$-gonal cross-sections by parallel planes. In conclusion, we note that by iterating this construction it is possible to obtain isoprisms containing polygonal cross-sections (lying in parallel planes) with $2^k n$ edges, for any $k\in \mathbb N$. 
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