SYMMETRIES OF FRIEZE ORNAMENTS IN \((n+1)\)-DIMENSIONAL SPACE

TIBERIU ROMAN

(Presented by Academician A. V. Shubnikov on 13 III 1962)

In the preceding paper \((^1)\), the symmetries of 4-dimensional frieze ornaments were defined and found. With the aid of the concept of antisymmetry, introduced by A. V. Shubnikov \((^2)\), frieze ornaments were represented geometrically in 3-dimensional space as two-color ornaments, and the structure of their groups of symmetries and antisymmetries was established. In the present paper the structure of the symmetry groups of \((n+1)\)-dimensional \((n \ge 1)\) frieze ornaments is studied; the number of these groups is found for all orders \(2^k\) \((k = 0, 1, 2, \ldots, n, n+1)\) with the aid of the concept of \(\alpha\)-symmetry of dimension \((n+1)\), which generalizes the concept of antisymmetry from \((^2)\).

Let transformations of \((n+1)\)-dimensional Euclidean real space be given by the relation \(y = Ax + a\), where \(A\) is a unitary matrix (i.e. \(AA^{+}=E\)) and \(a\) is an \((n+1)\)-dimensional vector. For such transformations we shall use the notation \((A,a)\). It is known that the multiplication of two such transformations is carried out according to the rule
\[ (A,a)\cdot(B,b)=(AB,Ab+a). \]

Definition 1. We shall call an elementary set a compact set \(\mathfrak m\) of points of \((n+1)\)-dimensional space \((n \ge 1)\), satisfying the following axioms: 1) \(\mathfrak m \subset S\), where \(S\) is a sphere of the given radius; 2) \(O \in \mathfrak m\), where \(O\) is the origin \((0,0,\ldots,0)\); 3) \(\mathfrak m \cap H_1\) admits no unitary transformations into itself except diagonal transformations \((H_1\) is the hyperplane \(x_1=0)\); 4) \(\mathfrak m\) admits no transformation into itself of the form \((E,a)\), where
\[ a= \begin{pmatrix} a\\ 0\\ \vdots\\ 0 \end{pmatrix}, \qquad a\ne 0. \]

Definition 2. A frieze ornament is a point set of \((n+1)\)-dimensional space, obtained after transforming an elementary set \(\mathfrak m\) by the group \(C_{\infty}\), consisting of translations \((E,kt)\), where \(k\) is an integer,
\[ t= \begin{pmatrix} 1\\ 0\\ \vdots\\ 0 \end{pmatrix}. \]

Definition 3. We shall call symmetries of a frieze ornament transformations of the form \((A,a)\) that map the ornament onto itself. Two symmetries \((A,a)\) and \((B,b)\) are called geometrically equivalent if \(A=B;\ a=b+kt\) \((k\) an integer) or
\[ (E,-{}^{1}/_{4}t)(A,a)(E,{}^{1}/_{4}t)=(B,b+kt). \]

Definition 4. Rotations of a frieze ornament are its symmetries of the form \((A,0)\), and rotations with subsequent translations are symmetries of the form \((A,{}^{1}/_{2}t)\).

The following theorems are easily proved:

Theorem 1. The matrices of rotations are diagonal.

Corollary. There are altogether
\[ C_{n+1}^{p}=(n+1)!/p!(n-p+1)! \]
rotation matrices that contain \(p\) \((1 \le p \le n)\) elements equal to \(-1\); there is only one matrix \(I\), all nonzero elements of which are equal to \(-1\).

By \(S_p^i\) \((i=1,2,\ldots,C_{n+1}^p)\) we shall denote one of the matrices that has \(p\) \((1\leq p\leq n)\) entries equal to \(-1\); in particular, we introduce the notation

\[ S_1^{\,n+1}= \begin{pmatrix} 0\ldots 0 & 0\\ 0 & 1\ldots 0 & 0\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ 0 & 0\ldots 1 & 0\\ 0 & 0\ldots 0 & -1 \end{pmatrix} =\sigma. \]

Theorem 2. In every symmetry group of a frieze ornament the group \(C_\infty\) is a normal divisor.

Theorem 3. All geometrically inequivalent symmetries of frieze ornaments are: the identity transformation \(1=(E,0)\); rotations \((S_p^i,0)\), where \(p=1,2,\ldots,n;\ i=1,2,\ldots,C_{n+1}^p\); rotations with subsequent translations \((S_p^j,{}^{1}/_{2}t)\), where \(p=1,2,\ldots,n;\ j=1,2,\ldots,C_n^p\) (since the element in the upper left corner cannot be equal to \(-1\)); inversion \((I,0)\).

These definitions and theorems generalize the first definitions and theorems 1—3 from \((^{1})\).

To determine the symmetry groups of \((n+1)\)-dimensional frieze ornaments, new definitions and theorems are given that extend the last definition and theorem 5 from \((^{1})\) and the theorem from \((^{5})\):

Definition 5. An \((n+1)\)-dimensional \(\alpha\)-symmetry is a symmetry of an \((n+1)\)-dimensional frieze ornament for which the element in the lower right corner of the matrix \(A\) is equal to \(-1\). The other symmetries of an \((n+1)\)-dimensional frieze ornament will be called \(\beta\)-symmetries.

Corollary 1. An \((n+1)\)-dimensional \(\alpha\)-symmetry may be regarded as a symmetry of an \(n\)-dimensional frieze ornament with a subsequent change of the signs of points.

Corollary 2. The \(\beta\)-symmetries of an \((n+1)\)-dimensional frieze ornament may be regarded as symmetries of an \(n\)-dimensional frieze ornament.

Definition 6. The transformation \(\bar{1}=(\sigma,0)\) is called an anti-identity transformation; the transformation \(\bar{A}=(\sigma,{}^{1}/_{2}t)\) is called an antitranslation by \({}^{1}/_{2}t\).

Definition 7. A set of \(\beta\)-symmetries of an \((n+1)\)-dimensional frieze ornament which is a group is called a generating symmetry group of \(n\)-dimensional two-color frieze ornaments (briefly: a generating group \(n_2\)).

The direct product of a generating group \(n_2\) with the group \(\{1,\bar{1}\}\) is called a neutral symmetry group of \(n\)-dimensional two-color frieze ornaments (briefly: a neutral group \(n\)).

The direct product of a generating group \(n_2\) with the group \(\{1,\bar{A}\}\) is called a pseudoneutral symmetry group of \(n\)-dimensional two-color frieze ornaments (briefly: a pseudoneutral group \(n_2\)).

The other symmetry groups of \((n+1)\)-dimensional frieze ornaments are called mixed symmetry groups of \(n\)-dimensional two-color frieze ornaments (briefly: mixed groups \(n_2\)).

Theorem 4. The symmetry groups of \((n+1)\)-dimensional frieze ornaments are: generating groups \(n_2\); neutral groups \(n\); pseudoneutral groups \(n_2\); mixed groups \(n_2\).

Mixed groups \(n_2\) are obtained as follows: the generating groups \(n_2\) are decomposed into right (or left) cosets with respect to all their subgroups of index 2, and in all classes that do not contain the identity of the group, \(\beta\)-symmetries are replaced by the corresponding \(\alpha\)-symmetries.

Now one can establish the main theorem of the present paper, which gives the structure of the symmetry groups of \((n+1)\)-dimensional frieze ornaments and the number \((2^k)_{n+1}\) of all groups of order \(2^k\) of geometrically inequivalent symmetries.

Theorem 5. \((n+1)\)-dimensional frieze ornaments admit groups of geometrically inequivalent symmetries of order \(2^k\) \((k=0,1,2,\ldots,\ldots,n,n+1)\); their number is:

\[ (2^0)_{n+1}=1;\quad (2^1)_{n+1}=3\cdot 2^n-2; \]

\[ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \]

\[ (2^k)_{n+1}= \frac{(2^n-1)(2^{n-1}-1)\ldots(2^{\,n-(k-2)}-1)} {(2^2-1)(2^3-1)\ldots(2^k-1)} \left[(2^k+1)2^n-2^k\right], \tag{1} \]

where \(k=2,3,\ldots,n,n+1\).

Indeed, rotations, rotations followed by translations, and inversions (Theorem 3) are elements of order 2, since \((S_p^i,0)\cdot(S_p^i,0)=(E,0)\); \((I,0)(I,0)=(E,0)\), while the symmetry \((S_p^i,\tfrac12 t)\,(S_p^i,\tfrac12 t)=(E,t)\) is geometrically equivalent to \((E,0)\).

From elements of the second order one can form only groups of order \(2^k\) of type \((2,2,\ldots,2)\).

\[ \underbrace{\hspace{2.8cm}}_{k} \]

It is obvious that there is only one group of order \(1=2^0\): \((E,0)\). To determine the number of groups of geometrically inequivalent symmetries of the second order, note that they have the form \(\{1,U\}\), where \(U\) is an inversion, a rotation, or a rotation followed by a translation. The number of rotations is
\(C_{n+1}^1+C_{n+1}^2+\cdots+C_{n+1}^n=2^{n+1}-2\) (by the corollary to Theorem 1); the number of rotations followed by a translation is
\(C_n^1+C_n^2+\cdots+C_n^n=2^n-1\). Hence it follows that the number of groups of the second order is given by the formula

\[ (2^{n+1}-2)+(2^n-1)+1=3\cdot 2^n-2. \]

To determine the number \((2^k)_{n+1}\) of groups of geometrically inequivalent symmetries of order \(2^k\) \((k=2,3,\ldots,n+1)\) of \((n+1)\)-dimensional frieze ornaments, Theorem 4 is applied: the number of generating groups \(n_2\), geometrically inequivalent, of order \(2^k\) is equal to \((2^k)_n\); the number of neutral groups \(n\), geometrically inequivalent, of order \(2^k\) is equal to \((2^{k-1})_n\), since these groups are direct products of generating groups \(n_2\) (of order \(2^{k-1}\)) with the group \(\{1,I\}\); the number of pseudoneutral groups \(n_2\), geometrically inequivalent, of order \(2^k\) is equal to \((2^{k-1})_n\), since they are direct products of generating groups \(n_2\) (of order \(2^{k-1}\)) with the group \(\{1,\bar A\}\); the number of mixed groups \(n_2\), geometrically inequivalent, of order \(2^k\) is equal to \((2^k-1)(2^k)_n\), since each of the \((2^k)_n\) generating groups \(n_2\) is decomposed into 2 right (or left) cosets with respect to \(2^k-1\) subgroups of index 2 (one subgroup has first order), one of these two classes does not contain the identity of the group. Consequently, the recurrence relation is established

\[ (2^k)_{n+1}=2^k(2^k)_n+2(2^{k-1})_n. \tag{2} \]

For \(k=2\) this relation makes it possible to obtain the particular case of (1):

\[ (2^2)_{n+1}=\frac{2^n-1}{2^2-1}\left[(2^2+1)2^n-2^2\right]. \]

Formula (1) is proved by a twofold application of the method of complete induction. It is valid for \(k=2\); suppose that it is also satisfied for \(k=p\); this makes it possible to compute \((2^p)_{n+1}\) and \((2^p)_n\). The rela-

relation (2) for \(k=p+1\) can be written in the form

\[ (2^{p+1})_{n+1}=2^{p+1}(2^{p+1})_n+ 2\frac{(2^{n-1}-1)\cdots(2^{n-p+1}-1)} {(2^2-1)\cdots(2^p-1)} \left[(2^p+1)2^{n-1}-2^p\right]. \tag{3} \]

Since \((2^q)_2=0,\quad (2^q)_3=0\ (q=4,5,\ldots)\) (see \((3)\) or \((4)\)), it follows from (2) that

\[ (2^{n+h})_n=0.\qquad (h=1,2,\ldots). \tag{4} \]

Replacing \(k\) by \(k+1\) in (2), we obtain

\[ (2^{k+1})_{n+1}=2^{k+1}(2^{k+1})_n+2(2^k)_n. \tag{2'} \]

Formula (1), which was assumed to be valid for \(k=p\), gives
\((2^p)_p=2^{p-1}\), whence, for \(k=p\) and \(n=p\), \((2')\) leads to
\((2^{p+1})_{p+1}=2^{p+1}(2^{p+1})_p+2(2^p)_p\), and, taking (4) into account,
\((2^{p+1})_{p+1}=2^p\).

Consequently, relation (1) is verified for \(k=p+1\) and \(n=p\) (for \(n<p\) it is also valid, since, by (4), \((2^{p+1})_{n+1}=0\)). Relation (3) makes it possible to compute \((2^{p+1})_{p+2}\):

\[ (2^{p+1})_{p+2} = 2^{p+1}(2^{p+1})_{p+1} + 2\frac{(2^p-1)\cdots(2^2-1)} {(2^2-1)\cdots(2^p-1)} \left[(2^p+1)2^p-2^p\right] = \]

\[ =2^{p+1}\cdot 2^p+2\cdot 2^{2p}=2^{2(p+1)}, \]

whence it is clear that (1) is verified for \(k=p+1\) and \(n=p+1\).

Assume that (1) is valid for \(k=p+1\) and \(n=m>p\). Relation (2) gives

\[ (2^{p+1})_{m+1}=2^{p+1}(2^{p+1})_m+2(2^p)_m. \tag{5} \]

But \((2^{p+1})_m\) and \((2^p)_m\) can be computed by means of (1) on the basis of the assumptions made above. Relation (5) makes it possible to compute \((2^{p+1})_{m+1}\), which agrees with what (1) gives. Consequently, (1) is valid for \(k=p+1\) and all \(n\). Since (1) is valid for \(k=2\) and arbitrary \(n\), and it has been proved that it is valid for \(k=p+1\) and all \(n\), Theorem 5 is completely proved.

Remark. The number of groups of geometrically inequivalent symmetries, of order \(2^k\), of \((n+1)\)-dimensional frieze ornaments is given by the following table, from which one can also compute the previously known values (see \((3)\) or \((4)\) and \((1)\)) for 2-, 3-, and 4-dimensional spaces:

\[ (2^0)_{n+1}=1;\qquad (2^1)_{n+1}=3\cdot 2^n-2;\qquad (2^2)_{n+1}=\frac{2^n-1}{2^2-1}(5\cdot 2^n-4); \]

\[ (2^3)_{n+1}= \frac{(2^n-1)(2^{n-1}-1)} {(2^2-1)(2^3-1)} (9\cdot 2^n-8); \]

\[ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \]

\[ (2^k)_{n+1}= \frac{(2^n-1)\cdots(2^{n-k+2}-1)} {(2^2-1)\cdots(2^k-1)} \left[(2^k+1)2^n-2^k\right]; \]

\[ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \]

\[ (2^{n-1})_{n+1}=\frac{2^{2n}-1}{2^2-1}\,2^{n-1};\qquad (2^n)_{n+1}=2^{2n};\qquad (2^{n+1})_{n+1}=2^n. \]

Bucharest,
Romania

Received
7 III 1962

REFERENCES

1. T. Roman, DAN, 128, No. 6 (1959).
2. A. V. Shubnikov, Symmetry and Antisymmetry of Finite Figures, Moscow, 1951.
3. A. Speiser, Die Theorie der Gruppen von endlicher Ordnung, Berlin, 1927, 1937, 1954.
4. T. Roman, An. Univ. C. I. Parhon, București, Ser. St. Nat., No. 16 (1957).
5. A. M. Zamorzaev, Kristallografiya, 2, issue 1 (1957).