Tomb towers and minarets: analysis of symmetries and geometries of Iranian geometrical ornaments of the Seljuq era. Pictorial requiem for the Kharraqan towers

Ornamental adornment of the Kharraqan tomb towers, the most outstanding funeral monuments of the Seljuk era in NW Iran, and those of four best-preserved Seljuq brick minarets in northern Iran, documents the artistic canon of the pre-glaze stage of Iranian Islamic architecture. Despite some later interruptions, these monuments and their plain-brick ornaments, as well as the ‘virtually interlaced’ brick ornaments, stand at the beginnings of a rich development that led to the Safavid architecture of Iran. Besides documentation and study of the geometric character of early Islamic art, which was based on limited technical resources, this study offers insight into symmetry concepts developed at this stage of art and architecture development. This is the last and most complete study of the Kharraqan towers performed before their overwhelming destruction in the 2002 earthquake.


Introduction
The Seljuq Islamic architecture of 'pre-glaze age' in Iran and immediately adjacent regions is constructed primarily of bricks, and not of worked stone which was used in Seljuq Turkey. It not only has a unique art value but also represents one of the peaks of application of symmetry to ornamental art. This study investigates ornamental adornment of the Kharraqan tomb towers that count among the most outstanding funeral monuments of their time, and of four best-preserved Seljuq brick minarets in Iran. These monuments stand at the beginnings of a rich development that resulted in such achievements as the quasi-periodic patterns of the Gunbad-e-Kabud (Blue Tower) at Maragha, W Iran (cf. Makovicky 1992Makovicky , 2008Makovicky , 2016Lu and Steinhardt 2007). In parallel to the results and conclusions concerning the geometric character of early Islamic art, this study offers insight into symmetry concepts developed (or those still absent) at this stage of art and architecture development.
Studies undertaken by these authors differ-they range from the analysis of inscriptions, and of the architectural details, association with metaphysical concepts, to studies of ornament geometry. Still, they left enough space for a thorough symmetry and underlying geometry analyses, which are the main topic of the current paper. I agree with Bier (2002) that the ornament designer wished to explore the mathematical properties of the two-dimensional space, doing it with exuberant expressiveness. I see, however, more of a professional vigor supported by the wishes of the This paper belongs to the topical collection "Quasicrystals: State of the art and outlooks" originated from an international conference organized by the Accademia dei Lincei, held in Rome on November 18, 2022 in the frame of the 2022 International Year of Mineralogy.
order-giver to represent best his clan, instead of a metaphysical approach and hidden meaning in the choice of ornaments used, which she suggested.

Historical frame
Arab domination of the eastern Islamic lands came to its end in 945 when Arab caliphs had to surrender their worldly authority to the military commanders of Persian origin. For about one century, this area was ruled by dynasties of Persian origin, especially by Buyids. Later, power went to assimilated Turcoman Ghaznavids (Hillebrand 1999). For the architectural history of Iran, however, the decisive moment is that in 1037, when another Turcoman group, the Seljuqs (also spelled as Saljuqs, Seljuks) conquered Merv, then the central and western Persia, occupying Esfahan in 1051, and Baghdad in 1055. In the last quarter of eleventh century, they ruled over the entire Western Asia (Rice 1991;Bonner 2017).
The Seljuq era brought about the first extensive blossoming of Iranian art (Hillebrand 1999). The Great Seljuqs of Iran were lavish building patrons-mosques, madrasahs, minarets, small mausolea (imamzadas), caravanserais, and the typical gumbads (tomb towers). The same high level of activity was present in all other branches of applied artsceramics, metalwork, book painting, etc.
This activity, however, came to an abrupt halt by Mongol invasions in 1219 and the following years. True renaissance of Iranian arts comes only about 90 years later, under the Ilkhanid ruler Ghazan Khan (1295-1304) (Hillebrand 1999).

Materials and their use
Owing to the level of technological development of the day, much of the Seljuq architecture is adorned by unglazed brick ornaments, in a style called 'brick style' or hezar baf (thousand weavings) (Clevenot and Degeorge 2000). This technique created ornamental panels or entire walls from variously laid and variously oriented bricks, which were occasionally halved, quartered, cut, or beveled to angles other than 90°. Sometimes, they were combined with terracotta stamped with designs or with stucco. The latter, however, was used amply for interiors. These techniques were mostly inherited from older periods (as witnessed, e.g., by the Tarik Khana Minaret, about 1027) but they were brought to perfection by the builders of the Seljuq period. Monumental inscriptions adorn these buildings; the entire 'brick style' requires the play of strong light and shadows produced by the slowly moving sun in the mostly clear Iranian skies.
The pale-to-azure blue glaze appears later in the Seljuq period, e.g., as an epigraphic pale-blue band on the brickstyle minaret of the Jami Mosque in Damghan (about 1058), colored bands and inscriptions on Masjid-i-Jami in Qazvin (1113) or a geometrical band of glazed bricks on the minaret of the Kalyan Mosque (1127). Extensive use of glazed brickwork is found on the Blue Tomb of Maragha (1147) (Makovicky 2008;Makovicky and Ghari 2018) but its blossoming is connected with the Ilkhanid architecture. In the Timurid period (1370-1506) the glazed bricks/tile mosaics took over the role of architectural ornament (Clevenot and Degeorge 2000), leading to virtual extinction of the 'brick style' (Makovicky 1989).
For the studies of ornamental art, many authors use combination of fundamental geometry and subjective impressions; the fullest representation of such description is that by Bier (2002, part 3.1). There exists an exact language, however, which deals with a distribution of individual motifs and their combinations-it describes symmetries of zero-dimensional, as well as of (periodic) uni-and two-dimensional patterns.
The symmetry language was created and defined mostly for physical sciences, but it fits perfectly to the study and classification of periodic ornaments. Moreover, we shall see that the architect was in some cases thinking it terms of two-sided two-dimensional patterns (one visible side and one symmetryrelated, usually identical, inaccessible side), which are a result of illusionary interlace (Bier 2002)-the language of symmetry manages this problem with ease.
The present study is based on own observations and photographic evidence collected in Iran in 2002.
Unfortunately, this contribution is a pictorial requiem for the beautiful ornamental towers of Kharraqan. Just three weeks after my visit in 2002, on July 22nd, a powerful earthquake (Walker et al. 2005; Wikipedia anonymous) with epicenter located 59 km SW of Abhar (SW of Qazvin), and with magnitude calculated to 6.5 or more at the epicenter, irreparably destroyed the towers (which otherwise survived centuries). It caused one of them to sink into the ruins of its lower portion and split the other into upward-raging ruins (Jean-Marc Castera, private communication); all of this destroyed many ornaments on the walls. Locally, it was the strongest earthquake in about 900 years. Thus, this contribution is the last one describing the well-preserved and professionally restored situation before the demise. Although still described in the present tense in this paper, the optimal condition of these towers, described here, is a matter of the past.

The Kharraqan tomb towers
Perhaps the most spectacular object of early Seljuk architecture are two Seljuq tomb towers at Kharraqan in Western Iran (Fig. 1), built by one and the same architect for Turkoman chieftains in the years 1067 and 1086, respectively. These tomb towers lie about 120 km to the southwest of Qazvin, about 30 km up a broad lonely valley stretching to the west from the village of Abgerme, which itself is situated on the Takestan-Razan Road. The locality is less than one kilometer to the west of a small, poor village, known in Azari as Hesar Armani, later Hesar Valiars. Local stories describe these towers as mausolea for an assassinated holy man and his sister who shared his fate or as a place of living of a Seljuq governor. In spring, before harvest, people from surrounding villages used to gather for a celebration honoring the legendary assassinated imam with a sacrifice of a sheep.
The inscription on the W tomb tower proclaims Muhammad ibn Makki al-Zanjani as the architect, although the names of clients are less certain. The tower tombs are octagonal prisms, about 13 m tall, each terminated with a brick dome, which apparently had been restored. Both towers have powerful semicircular buttresses attached to all corners. They were built of buff-fired bricks, indicating calciumenriched clay compositions.
The ornamental fields of the two towers differ in their layout. The eastern tower has large arched blind 'windows' which take up most of the flat wall space (Fig. 1). These 'windows' are surmounted by broad horizontal friezes that have an inscription at their base. Simple cmm brickwork fills the triangular spaces left between these elements. Buttresses are covered by ornamental brickwork as well, and flanked by diminutive, ornamented pillars; whereas, the upper, arched frames of ornamental 'blind windows' are left unadorned. Lower portions of the walls and buttresses lost their ornamental coating to humidity and very little is left of the original dome as well.
The western tower ( Fig. 1) has 'blind windows' divided into tall, rectangular lower portions and arched upper portions. They are separated by a row of three small ornamental panels surmounted by trefoil arches. Again, the ornamental friezes with an inscription at the base surmount the walls. Buttresses carry rich ornamentation. Small pilasters flank the flat, ornamented walls up to the horizontal dividing line. Top arches are partly ornamented by a linear ornament, partly left unadorned. Some buttresses and the bottom portions of the walls lost their ornamental coating.
In the interior of the W tower, a mihrab wall is preserved. It is a flat panel of two-dimensional ornament, surmounted by a smaller panel under a pointed trefoil arch. Both towers underwent expertly restoration without forcing new additions upon the original brickwork.
Although coming to Western attention at a rather late date (Stronach and Young 1966), panoramatic picture of Kharraqan towers and less frequent pictures of one or two of their ornamental panels can be found in many publications on Islamic art of Iran, e.g., Ettinghausen and Grabar (2001), actually too many to quote. A more comprehensive selection of the latter is in Makovicky (2016), although it is much smaller than the true number of extant two-dimensional ornaments. The current publication attempts to remedy the situation, especially because many depicted patterns succumbed to the earthquake. Among the specialized publications devoted to the Kharraqan towers, we might mention Stern (1966) who deals with the inscriptions, Stronach and Young (1966) dealing with the tower architecture, and Bier (2002) who suggests that the geometric patterns on the towers have metaphysical meaning and combine to form a programmatic cycle of meanings.

Seljuk minarets
Towns on the old commercial route (Silk Road) along the southern limits of the Elburz Mountains, and those encircling the vast Dasht-e-Kavir desert, were dotted with mosques that boasted with exquisite minarets adorned with complex brick ornamentation. In some cases, these minarets are the last vestiges of the old building, the mosque itself having succumbed to numerous alterations.
I studied the minaret of the Jami' Mosque at Saveh, northwest Iran (built in 1110-1111 according to Hutt and Har-row1977) at first-hand, as well as those of the Jami' Mosque in Semnan (eleventh century), Jami' Mosque in Damghan (about 1058), and the Tarik Khana minaret of the Great Mosque in Damghan (probably pre-Seljuq, 1027). All three are situated in northern Iran. Books by Hutt and Harrow (1977), Michell (1978), Stierlin (2002), and Clévenot and Degeorge (2000) offer additional, although rather sparse material concerning these objects.

Methods and problems
As mentioned above, two-dimensional crystallography offers unambiguous alternatives to the usual classification of ornaments as 'complicated', 'dynamic', 'static', etc., by identifying individual symmetry elements in the ornamental patterns studied, and analyzing their combinations, known as the point groups (supplement data: Fig. S1), plane groups, layer groups and 1-dimensional frieze groups of symmetry. We can obtain overall, or partial statistics of plane/frieze group symmetries used by the artisans/architects from such studies. Practical experience reveals that friezes can be divided into (a) possible cut-outs of 2D patterns observed in the region and (b) the truly one-dimensional patterns; it was done by Makovicky and Fenoll Hach-Alí(1997b) for the Andalusian Islamic art. Especially for the Kharraqan case, the illustrations included in the text, will be supplemented by a supplementary set, indicated by 'S' and a running number of supplementary illustrations.
Often, several different patterns (in one region) have the same plane-group symmetry. For their differentiation and a more detailed classification, special and general positions of individual elements (motifs) of the pattern (so-called Wyckoff sets) placed in the unit mesh are studied (Fig. 2). This means that the motif elements (tesserae) can either be placed on rotation axes or they can straddle reflection planes of symmetry (these are the special positions) or they can be situated in spaces between these symmetry operators (in socalled general positions). This determines both their shapes and the multiplicity with which they occur in the pattern. There may be interpretational problems with application of this principle to those patterns which are composed of endlessly interlaced ribs: should we consider the recessed finite volumes between ribs as 'tiles' or should we take the protruding continuous lines as the basic elements? In such cases, we must select elements (interpretations) which appear to be closest to the understanding of the original masters, e.g., by virtue of their frequent and repeated use in the region.
Based on the types of positions occupied by the different elements of the pattern, and on the number of times individual position types are occupied by distinct elements, patterns can be divided into three categories. Simple patterns have only one set occupied and only once, intermediate patterns have 2-3 Wyckoff sets occupied or a certain set occupied more than once, and the complex patterns, in which many elements combine, usually display a multiple occupation of certain positional sets.
It is not the purpose of this paper to re-explain and tabulate groups of symmetry; the purpose is to use them. The plane groups of symmetry can be found illustrated and geometrically described in the International Tables   for Crystallography published by the IUCr's Commission  on International Tables, volume A (2016) (quoted as IT below, available online), as well as in the reference books Their symbols suggest the local tile symmetry and shape, constructed as coalesced groups from the small right-angle triangles, which were used as separated only for the general position g. Different position types shown here can coexist in one pattern by Washburn and Crowe (1998 and re-prints), Abbas and Salman (2007) and Makovicky (2016). A copy of a table/ illustration of these groups in your hand is recommendable for fully appreciating the following chapters of the present paper. The purpose of our contribution is to use the symmetry groups in a critical way, as it was done by, e.g., Makovicky and Fenoll (1997a). The frequently occurring question of pseudosymmetry (when metrics/symmetry of the underlying geometric scheme is higher than that of the resulting pattern), or of the excess local symmetry, will be addressed as well.
Islamic artists were the greatest masters of two-sided layer symmetry, and the interlaced patterns, which are based on this kind of symmetry, are a frequent adornment of the important localities of Islamic art. Ignoring the true layer symmetry of these patterns, and replacing them automatically by plane-group symmetry, is rather frequent in literature, but it distorts the understanding of given art style.
Very interesting is the profusion of interlaced patterns already in early Seljuk art, including the studied localities. With the technical limitations of the patterns created by brick arrangement, the Seljuks artists mimicked (mostly in an error-free fashion) the interlacing by carefully arranging the component bricks in such a way that the pattern creates an impression of interlacing with each string alternatively 'going over' and 'going under' the strings it meets. The best source of 'layer groups of symmetry' (Fig. 3), needed for this portion of study, are the International Tables for  Crystallography, published by the IUCr's Commission on  International Tables, volume E (2010), whereas instructive examples are in, e.g., Makovicky (2016).
There exists another classification scheme for patterns, which is the so-called structural classification. This classification separates them into families, which contain the same or very similar elements and element-combination principles, but their members differ by well defined, incremental changes. For example, certain segments of the pattern grow incrementally, and the consecutive members of the family differ by a number of brick courses that build these portions. At the same time, the other portions, situated between the former ones, remain unchanged. These are so-called homologous series constructed by means of element accretion. Alternatively, one can modify the pattern by intercalation of additional ornamental elements into the original pattern ('baroquization' of the pattern or creating pattern-complication series of gradually developing patterns). Further structural types of patterns and series are the expansion-reduction series, omission derivatives, elementsubstitution series, partial-overlap derivatives, vortex series, etc., which were sometimes developed by the artists (Makovicky 1989; Makovicky and Fenoll 1997a); for similar pattern categories, see Roe (1980). Although the present study offers only a limited set of examples for these categories, we shall mention them whenever applicable.

The patterns
The one-dimensional and two-dimensional brick patterns of the Kharraqan tomb towers and of the Seljuq minarets can be divided into three categories: (a) Flat-surface patterns created by orientation of brickstacking ('flat-brick patterns'), (b) Patterns created by protruding ribs and recessed background, both materialized as brickwork ('rib patterns'), (c) Complex and very rich, 'loaded' rib patterns.  (2016) I shall describe the diverse ornamental patterns observed on the Kharraqan tomb towers as first. Patterns used on the walls of minarets display specific common features, and we shall handle them separately in the subsequent text.
The flat-brick-stacking patterns of the two towers ( Fig. 4) consist of standard 1:2:3 bricks, which were inserted in the body of the wall as a combination of 'horizontal' and 'vertical' orientations, i.e., as two orientations 90° apart. Sometimes the patterns were completed by adding the 1:1 square bricks to the standard ones. There are two practical interpretation problems connected with these patterns: (1) The large scale. They often exceed or nearly exceed the size of the panel, making determination of their periodicity and plane-group symmetry problematic. (2) The vertical versus horizontal orientation of the bricks, used to create a visually distinct pattern of shadows, was the principal tool of artistic expression. From a distance it results in a visual 'damask effect'. This arrangement invites dual interpretation of pattern symmetryeither an approach, which ignores the brick orientation and considers only the resulting shapes and disposition of the fields outlined by them (this approach will be used here), or an approach, which includes brick orientations into final interpretation, and results in reduced symmetry.
(3) The size of certain portions of these patterns can sometimes be varied by increments, adding another brick or a brick course (stripe) to them. In this way, we obtain the already mentioned homologous series of patterns.
All members of such a series are constructed using the same pattern principle.
Two principal observed types of flat-brick patterns are (a) patterns of (sometimes framed) squares with or without a small central cross which is formed by a standing brick and two square bricks which flank it (Figs. 4,5), and (b) patterns with larger framed crosses accompanied by smaller squares or frames (Figs. S2, S3, S5). Figure 5 shows two members of a homologous p4mm series built from ornamental squares, and situated on two adjacent buttresses. The higher member has two additional brick courses added to the edge of the diamond-oriented squares, which characterize both members of the series. Centers of these squares define the vertices of the square unit mesh and house fourfold axes of rotation symmetry (in the interpretation sub (a) above; the other fourfold rotation axis lies in the intersection of single-brick courses  Two members of the homologous series formed by a squarebased p4mm pattern, with 7 and 11 tiers counted along the vertical diagonal of the unit square, covering two separate buttresses. A framed p4mm pattern of squares and crosses covers the intervening flat panel. Buttresses flanking the S panel of the E tower separating the squares. Because of the brick orientation, a strong visual impression of cmm is present (this ambiguity was mentioned sub (2) above). The square diagonals in the two homologues are, respectively, 7 and 11 brick thicknesses long. The Damghan minaret has the same pattern with square diagonals 9 brick thicknesses in diameter, and with recessed central portions. Fig. S4 (in the supplementary set) and Fig. 4 display the squares framed by different frames, with symmetry p4mm (Fig. 4) but the patterns are too large to be resolved unambiguously.
The extant brick patterns with crosses have between six and nine bricks per entire arm of the cross. A well-preserved linear pattern p2mm is on a buttress shown in Fig. S3. The pattern with smallest crosses contains squares of two sizes (p4mm, Fig. 6) whereas those with larger crosses have double-crosses framed with a swastika-like configuration (p4gm, Fig. S5). Double crosses were also used in another large p4gm design. Finally, some large panels were interpreted only as point-group designs, with symmetries 4 and 4mm (Fig. S1).
The niches of the W tower are small and only rarely contain brick-stacking patterns. In Fig. 7a, they accommodate a simple p4gm pattern, composed of a combination of bricks with square bricks and an indefinite fragment of another, larger pattern, perhaps p2mg.
A rich spectrum of rib patterns with a contrast of raised bricks (viewed edge-on) and recessed fields, was used for tympana, panels, niches, and friezes of the tomb towers. Architects used it extensively also for the upper portions of minarets to create patterns visible from afar; whereas, the lower portions of the same minarets were usually covered by flat-brick-stacking patterns instead. In many patterns on the Kharraqan towers, the intricate brick layout, with one strand of (beveled) bricks crossing another, has been purposely designed as an illusionary-interlaced pattern and has to be interpreted as such, i.e., as a 'virtual' two-sided pattern and not as a plane pattern. This may not be true, however, for large-scale patterns on minarets, where the mutual positioning of bricks appears to be a technical matter.
The bricks employed in these patterns are flat bricks of different (exposed) length. When necessary, they were beveled on the 'wall-inserted' edges from the original 90° to (ideally) 67.5°, 60°, or 45°, to accommodate the orientation changes of adjacent bricks in the illustrated patterns.  Ornaments of this category, constructed using lowersymmetry plane groups, are largely absent. Only two 'diaper ornaments' were observed, adorning the lateral pilasters, which flank some panels, and were composed either of raised lozenges or of shaped mirror-symmetrical elements on a recessed background. They display plane groups cmm and cm, respectively (pilasters flanking several panel photographs). The cmm pattern of disjoint cubes is on pilasters in Fig. 7b.
The plane group p4 is present in two patterns. A 'curvilinear pattern' on a buttress (Fig. 8) has fourfold 'flowers' and squares, respectively, positioned on the two sets of fourfold axes. An impressive flat panel has swastikas alternating with small squares (Fig. 9); the latter have local symmetry in excess of that of the p4 plane group. The swastika pattern can also be interpreted using layer group p-4, when we examine the central bars of swastikas in detail: we find that the vertical segment 'virtually overlaps' the horizontal arm. This, however, might be a technical rather than esthetic factor: in this way, bricks can be better fixed to the brickwork.
Several interesting rib patterns obey plane group p4gm. A 'double-fork' ornament in a large tympanum (Fig. 10), and a related 'X'-based ornament on a panel (Fig. 11), lead to a 'pattern of pointed parentheses' (Fig. S6). The latter pattern, in all important features, is a dual of the X-based ornament. The forked elements in Figs. 10 and 11 are positioned in the points with local symmetry 2mm, with dot-like elements in two different positions. Finally, the fourfold axes in the p4gm pattern can be occupied by swastikas, in agreement with their point symmetry, and together they form small, indented rectangles on 2mm (Fig. 7b, central niche). This pattern is known from large panels elsewhere (Makovicky 1989, Fig. 34) as also is the 'maple leaf' p4gm pattern in Fig.  S7. We can see that the architect used the p4gm symmetry to construct several pattern types.
Although it is not immediately obvious, the apparent 'p4mm pattern' of interwoven brick ribs (Fig. 12) is composed of overlapping octagons, part of which comes from the outside of the pattern field. In spite of panel limitations, it is a 2D pattern, not a frieze pattern. The true, two-sided symmetry group is a layer group p422, and not p4mm; the virtual interweaving is flawless and intentional, with an ornamental purpose.
Bewildering by its complexity is a large-scale rib pattern in one of the tympana of the W tomb tower, composed of squares, pentagons and triangles with a dual net drawn over all of these elements (Fig. 13, see also Makovicky and Ghari 2018). The visual prominence of partitioned pentagons and rhombs obscures its true p4gm symmetry. The play of tilted  The tomb towers of Kharraqan display a profusion of beautiful hexagonal and trigonal rib patterns. The simplest hexagonal patterns are 'diaper' ornaments (e.g., Fig. 13, in the triangular side panels). Raised lozenges outline the recessed hexagonal stars in these patterns. Depending on the orientation of lozenges, both p6 and p6mm patterns were created. Several rib patterns with hexagonal symmetry were generated either by large overlapping hexagons centered on sixfold axes (Fig. 14) or, alternatively, by smaller overlapping hexagons centered on threefold axes (Fig. 15, top).
Nested hexagons create near-continuous parallel doublelines, which are combined with sixfold stars in the tympanum shown in the upper portion of Fig. 16 and in Fig. 14. A complex rib pattern of hexagons and zig-zag lines in Fig. 16 contains local 12.2.2 symmetries of the two-sided layer type, surrounded by 622 hexagons. It 'predicts', in a    In all these patterns, the plane group p6mm has been altered into a layer group p622. Only very rare interlacing errors were located, showing a high-quality work.
In another category of patterns, the pattern can be understood as constructed from, or completed by, zig-zag lines alone, with symmetry p6. Recessed spaces are based on threefold rotation axes, as triangular fields (Fig. 17), or they occur as S-shaped fields, which are based on twofold axes (Fig. 7a, the central niche). These two pattern types are duals of one another. A simple zig-zag line creates the p622 pattern in Fig. S9 whereas the construction of the doublehexagon version (Fig. S10) is more complex. These two versions differ also in the spacing of the underlying parallel construction lines: 1:1 spacings for Fig. S9 and approximate 3:1 for the S10 a pattern. Figure S10 b contains three different spacings in approximate ratio 4:2:1. Pattern in Fig.14 has alternating narrow and broader spacings as well. All this suggests that these simple underlying construction principles were already known and in use at the time of tower tomb construction.
The mihrab in the interior of the W tower hosts a repetition of the above-mentioned p6 pattern of S-shaped recessed fields in its upper portion. The principal panel of the mihrab wall, however, has a complicated p6 pattern composed of 'arrows' with side attachments. The arrows point outwards from the sites of sixfold axes. It is a two-sided layer pattern with intertwined ribs but its sixfold axes are polar and  there are no layer-reversing elements of symmetry present (Fig. 18).
There are two distinct trigonal patterns present: a 'palmate' p31m pattern with 'threefold snowflakes' on one set of threefold axes, and threefold swastikas on the other set of threefold axes (Fig. 19), and a pattern with a virtual 'interweaving' of double-strands that meet in one set of threefold axes of the layer group p-312 (Fig. 15, a large panel). The same pattern occurs on buttresses and, in a narrow form, as a frame of the adjacent tympanum (the 1D group is pg, now mostly preserved as p1 due to weathering). It is transitional to the flat-brick stacking.
The loaded rib patterns, each of them rich in a number of ornamental elements, form some of the most spectacular panels on the Kharraqan towers. The other patterns of this category are those applied to buttresses, in which the verticality of the motif becomes enhanced.
Most of these patterns were constructed with swastikas or recurved swastikas. However, nowadays they mostly contain 'lost swastikas', in which some of the short arm portions, probably originally fashioned out of stucco, disintegrated with time and in wet weather (Fig. S11). In two of these cases, remnants of the lost parts are recognizable in those panel portions, which remained protected by the arch of the tympanum. Thus, in the Fig. S11 pattern, the original p4gm symmetry was reduced to pgg by the loss of swastika arms and, in the weathered pattern, further reduced to pg because the small bricks that extend the diagonally oriented pair of bricks and are always situated at their upper end of the diagonal, were made bare and conspicuous. The pattern in Fig. 20 is more complicated because of the recurving swastikas, now reduced to sets of parallel, vertically oriented bricks. The original plane group p4gm has been reduced in the same way as in Fig.  S11. Figures 20 and S11 are two versions of one pattern type; mostly they differ only in details. For example, in the modification from Fig. S11, dots replace the small triangles, which are present in the version from Fig. 20.
Degraded swastikas characterize also the large-scale, originally hexagonal pattern of squares based on recurving swastikas, and arranged around sixfold stars and around a common vertex of three squares situated on threefold axes. The adjacent wreaths of swastikas share some of the swastikas. The underlying star-and-square pattern p6mm became reduced by insertion of swastikas to p6, but by further degradation of them by weathering, to final p2 (Fig. 21). The weathered version of virtually 'overlapping ribbons' acquired its own beauty. An expanded version of this pattern, in which the wreaths of swastikas around sixfold axes do not overlap, contains most of the stucco arms which were lost elsewhere (although they are present in a damaged form), and forms a nice p6 pattern (Fig. 13). We observed no two-sided reversal in these planar patterns.
Poorly preserved swastika arms occur also in a difficult, large-scale p4 pattern, which is the 3x3 superstructure of a simple swastika stacking which contains 2 x 2 blocks of swastikas (Fig. 22). The horizontal twofold axes, suggested by the framing (diagonally oriented) elements in the inter-block Fig. 18 The two-sided interlaced panel with a polar, non-reversing layer group p6. The recessed motif (with traces of blue paint) consists of arrows pointing outwards from the sixfold axes, adorned with side attachments. Mihrab inside the W tower; redrawn from a photograph  Fig. 24, in which the plane group p6mm has been altered quite purposely into a layer group p622. Its 'empty', not loaded version is the cartwheel pattern adorning a frieze of the E tower, with the same symmetry modification. It is shown in Fig. S13.
The 'arched' patterns are found mostly on corner buttresses; in their design, they follow the verticality of the buttress. Several cm patterns (Figs. 25, 26) and one pm buttress ornament (Figs. 26 and S14) were constructed with prefabricated floral embellishment. The swastikas mostly suffered the same weathering damage as was observed in the patterns already mentioned. The only cm pattern of this type, spread over a panel (Fig. 14) is very closely related to the cm pattern on buttresses and shares with it a number of motifs and overall design.

Pattern statistics
With their profusion of patterns, the Kharraqan towers invite a statistical treatment of pattern types and symmetry groups used. The eastern Kharraqan tower (Fig. S14) has

Fig. 20
The WSW panel of the W tower. In the original p4gm pattern, swastikas were on two sets of fourfold axes, whereas lying crosses were situated on twofold axes. Weathered to pgg; for details see text two patterns on each prism face; whereas, the western tower has six patterns on each face, except for the entrance panel. Corner buttresses and miniature pilasters flanking the panels carry additional patterns. For the western tower it ideally makes 45 panel patterns plus 16 buttress/pilaster patterns, i.e., 61 patterns. With two additional patterns in the interior, it makes ideally 63 patterns in all (without subtraction of the destroyed ones) on flat and curved surfaces. To them, we must add several one-dimensional patterns positioned on the tympanum arches. In this way, we reach potentially almost 70 patterns displayed on the W tower. The eastern tower has 32 patterns, what makes up to 36 patterns when the unidimensional patterns are included; from this we have to subtract some patterns which were destroyed. All top friezes on both towers are broad enough to allow interpretation as unmodified 2D patterns and not as 1D patterns. Overview of pattern distribution has been summarized in Table 1a and b, as well as in Figs. S15 a-d.
The visual role of patterns on flat surfaces and flat panels differs from that of the patterns enveloping cylindrical buttresses and pilasters. We shall treat them separately because the ornaments in these locations also differ in the choice of symmetry groups.

Panels on prism faces
When we exclude an occasional repetition of some patterns on a tower, the E tower has 5 distinct flat-brick (i.e., brickstacking) patterns versus 11 rib patterns whereas the W tower has 6 flat-brick patterns and 21 rib patterns. Hexagonal/trigonal patterns are absent among the flat-brick patterns; by their nature, they are primarily rectangular and quadratic patterns. Brick stacking patterns are absent in tympana.
Two kinds of statistical tabulations will be performed for the material from Kharraqan: (a) Frequency of plane groups of symmetry on all panels of a tower, including all repetitions of any pattern type with its plane group of symmetry as well as all repetitions of the plane group by means of different pattern types. This is the principle behind Table 1a and b. (b) Occurrence of distinct pattern types, discounting potential repetitions of some of the patterns on a tower, again expressed using their plane groups of symmetry. The results given in a graphical form are in Figs. S15a-d.
Both towers have a clear preponderance of the plane group p6mm, which, in a more detailed analysis, always is a layer group p622 in all cases. The western tower (Fig. S16) is distinguished by important occurrence of p4gm and p6 patterns, whereas for E tower p4mm (partly p422) is important, instead. The generally low frequency of p2, p4, cm, cmm, primary pgg, and pmg, and absence of p1, p3, (also of p6 on the eastern tower), pm, pmm, and p3m1 are conspicuous for panel ornaments in the Table 1.
Care must be exercised in the interpretation, as the p4gm pattern of swastikas has almost always been reduced to pgg in most of its area by weathering, except for the portions protected from the elements by the ornamental arch ledge. As already mentioned, the weather-unstable portions of swastikas 'withered away' during centuries of exposure. The long arms of the swastikas are composed of a long and a short brick in succession, reducing in some cases the fully weathered pattern to a plane group pg. Only the original plane groups were included in the count.
Repetition of patterns on towers is minimal, except for the niches where it is considerable, being one of the main reasons of differences between Table 1 and Fig. S15a-d. Search for correlations between plane groups of patterns on the same face/wall of the octagonal W tower gave no preferred combinations for panel-tympanum pattern pairs. Similar analysis of correlations between top friezes and large panels on both towers revealed that its results are determined by the absolute preponderance of p6mm (in fact, the layer group p622) in top friezes (Table 1a, b). The combination of any plane group, out of the entire spectrum of plane groups found on large panels or tympana, with a hexagonal 'p6mm' pattern on the frieze, is the usual scheme employed. Combination p6mm-p6mm between the panel and the frieze is rare on both towers, as also is occurrence of two quadratic groups together on the E tower. These statistics do not confirm the conclusions of Bier (2002) about pattern combinations.

Patterns on pilasters and corner buttresses
In the majority of cases, patterns on the curved surfaces of pilasters and corner buttresses can be unambiguously interpreted as cut-outs of 2D patterns; exceptions are several flatbrick, large-scale patterns, for which the extension itself is unclear. Only rarely, we deal with obvious 1D patterns, an example being the prominent brick-stacking pattern p2mm on two adjacent buttresses of the W tower (Fig. S16).
Frequency diagrams for this category (last two columns in Table 1 ,Figs. 15c,d) differ in several substantial features from those for the flat patterns which are on the faces of the octagonal prism, i.e., of the tomb tower. Principal plane groups of small pilasters are cm and cmm, representing primarily small-scale diaper-type patterns. For buttresses, especially the plane group p4mm occurs as flat-brick patterns. The rest of plane groups are present as one or two patterns each. Plane groups p1, p3,p6,p2mm,p3m1,pg,p2mg Fig. 26 Buttresses at the NW side of the E tower: arched patterns cm (foreground) and pm (rear). Left-hand side: a pilaster with a cm diaper pattern of pointed tiles surmounted by a p2 rim of a panel 1 3 are missing in this position on both towers. Hexagonal and trigonal patterns are extremely rare: we registered two p31m rib patterns on buttresses on the W tower, where they occur also on its tympanum, and found a diaper p6mm pattern on one of its pilasters.
The cmm patterns from pilasters were constructed mostly from bricks that are close to a square shape, with recessed plaster between them. Although they may approximate a square system, their geometry remains rectangular, i.e., cmm patterns. An impressive diaper cmm pattern composed of 'standing' slim lozenges is present on pilasters framing the NW side of the E tower.

Patterns used on frames of the panels
Several narrow frieze patterns form the frames of selected planar panels. Certain of them are fashioned of bricks, e.g., the p2 frieze framing the arch of the NE panel of the E tower, and surmounted by a p2mm frame consisting of bricks as well (Fig. 22). The complete combination occurs also on other faces of this tower and as an arch on the NEE panel of the W tower. Bead-like strings p2mm of shaped bricks and a damaged pg frieze derived from a trigonal pattern, framed by a p2mm stripe, are present on the W tower (Fig. 27).

Seljuq minarets
Except for the minaret in Saveh that has lower portions devoid of ornamentation because this fell victim to the elements, all the studied minarets have lower portions, two-four ornament tiers high, covered by flat-brick ornaments (Fig. 28). These are followed by upper portions, with between two and five tiers of rib ornaments (Figs. 29, S17, S18). Number of different positions on, or between, the Table 1 Distribution of ornamental patterns over the panels and buttresses of the western (a) and eastern (b) Kharraqan tower (situation before earthquake) a All instances of the p6mm plane group of symmetry are p622 on very close inspection. Those plane groups in the 'tympanum' and 'buttress' columns, which are indicated by asterisk, are layer groups on close inspection of their brick laying, as well. F indicates flat-brick patterns; the rest are rib patterns. Question marks indicate very large patterns, which exceed the panel area in size. b All cases of the plane group of symmetry quoted as p6mm are those of layer group p622 under more detailed inspection. The cases of p4mm* are those of layer group p422 in a similar way (details in the text). F = flat-brick patterns, all the rest are rib patterns To the contrary, the very impressive rib ornaments on these buildings are (very) complex patterns, with many recessed fields (some of them nested) which are positioned on a spectrum of general and special positions (Fig. 29). What makes the minaret ornaments distinct from the Kharraqan tomb towers is the decidedly limited choice of plane groups of symmetry. The bulk of cases are patterns in the plane group p4mm, with very rare occurrences of p4gm and of clear cmm (Figs. 28, 29, S17, S18). On the Semnan and Damghan minarets, several p4mm patterns are slightly compressed or extended along the vertical axis, yielding cmm when exact geometry is considered. For the cases, when such distortion is caused by brick/ mortar thickness and dimensions, we shall ignore these obviously pure technical modifications. No persuasive cases of interweaving were seen; the technical aspects of bricklaying and attachment seem to have determined the relations between adjacent bricks on these patterns. These patterns were designed to be contemplated from afar, from distances at which individual bricks coalesce into continuous ribs. Shadows created by the rib versus recess interplay play a decisive role.
Technically interesting is the limitation of ornamental ribs on the minaret walls to the vertical and horizontal ones, with rare cases of diagonal ribs, at 45° to the bulk of ribs present in the pattern. We did not observe any analogs to the rich choice of symmetries and geometries seen at Kharraqan. Details of brick arrangement reflect the technical solutions. In general, the superstructures are not simple multiples of small squares; sometimes half-squares were approximated.
Conspicuous is the repetition of these patterns over different locations. The flat-brick patterns with either elongate elements or full crosses inserted into framing occur on both minarets in Damghan, the latter one also at Semnan. Identical patterns of simple framed squares occur on the Tariq Khana minaret and the Semnan minaret. Two p4mm rib patterns on the two minarets built in Damghan are identical and one of them is repeated in Semnan. A pattern of large octagons 'interconnected' by elliptical links (Fig. 29) is common to Saveh ('upper' pattern, Fig. S18) and Masjid-e-Jameh, Damghan; its simplified versions are found on later minbars, etc. elsewhere in Iran. The bottom portion of the Damghan minaret is conical and the (possibly intended) p4mm flat-brick pattern becomes split and modified in its lower half. At the top of this portion, a repetition of nine 'standing' bricks and 14 lying bricks results in the cmm pattern with a separation of horizontal zig-zag strips. In the conical bottom portions, we observe a growing misfit of strips. In the cylindrical top portion, a 16 × 16 brick pattern creates large squares, spanning seven bricks, and is modulated by empty squares and braided fragments; plane-group symmetry p4mm. Graphic illustration of this pattern is in Makovicky (1989), (his Fig. 23)

3
The illustrated 'lower' Saveh pattern (Fig. S17) can be interpreted in two ways: (a) an intercalation derivative of a p4mm pattern in which the double-Y elements were first split and then doubled and separated from one another by a strip of short braided elements, or by the adjacent broader patch of fourfold groups, or (b) as a tetragonal primitive arrangement of octagons (which contain infill of 4 crosses) separated by 'collapsed' strips in which two octagon fragments overlap and enclose a braided remnant of the previous infills (Fig. S17). Both interpretations are possible, and both represent truly outstanding geometric creativity.
The just described pattern distribution can be extended to other Iranian minarets. The Barsiyan Mosque Minaret (1097) has a flat-brick pattern with squares 7 brick courses in diameter and crosses inserted in the framing, the Malik Mosque Minaret, Kerman (11 century), has squares 9 crosses broad, and simple, elongate inserts in framing, whereas the Shrineof-Bayazid Minaret at Bistam (probably 1120) is topped by two p4mm rib patterns, one of them a simplified version of the Damghan rib patterns (Hutt and Harrow 1977).

Conclusions
The planar ornaments on the faces of the octagonal tomb towers at Kharraqan show about equal numbers of tetragonal (square) patterns and hexagonal patterns, a phenomenon a bit unusual for ornaments constructed from bricks. For example, such ornaments from the times of Ummayad Emirate of Cordoba, Spain, contain substantial proportion of tetragonal patterns, besides the rectangular symmetry groups with twofold axes (Makovicky and Fenol Hach-Alí 1997a) but no hexagonal patterns. The statistic of plane groups from Kharraqan resembles rather closely that of the collection of Islamic ornaments by Bourgoin (1973), i.e., it looks like the average distribution of plane groups of symmetry, independent of the medium used. All 'p6mm' patterns belong to a two-sided layer group p622 in 'virtual' presentation. Buttresses of the younger, W tower have similar distribution, except for a high frequency of the cm designs. Buttresses of the E tower appear more influenced by the construction material. They lack three-and sixfold axes and resemble the Cordoban statistics.
Erosion of certain small elements of ornaments may have altered the overall plane-group symmetry and careful analysis of the entire pattern may be required to re-establish its original character (e.g., from pgg back to p4gm). Slight affine distortions, especially those of complex tetragonal patterns obviously were unintentional, a construction problem, and should be ignored. There exist exceptions, however, e.g., the minaret of the Jami'Mosque in Gurgan, Iran (12th century) on which the original p4 group has been affinely distorted to p2, with the original diagonals of a square altered from the ratio 1:1 to 1:2 (Makovicky 1989, his Fig. 18).
Except for flat-brick patterns of framed squares, among which cases with 5, 7, and 9 brick courses across a square occur, structural homologous series are rare in the sampled material, and limited to a few pattern pairs. Baroquisation of patterns by intercalation of additional elements is occasionally present.
The very narrow choice of plane groups of symmetry applied to the minarets studied, and the repetition, on different minarets in the studied region, of complex rib patterns that require complicated schemes of bricklaying, points towards one and the same workshop or to a tightly followed tradition with a sequence of pupils turned masters.