paper
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2007
paper:2022-06-06-prabros-bridges2007-203-pdf-6211a5

Frieze Patterns of the Alhambra

ByB. Lynn BodnerMonmouth University
Christopher Alexander's 15 propertiesGeometric structure in neural representationsAlhambra17 wallpaper groupsInternational Symbol Notation (pxyz)AlicatadoSeven frieze pattern groupscalligraphic inscriptionCounterchange SymmetryFrieze PatternsGlazed TileGlide-Reflection Symmetryhorizontal mirrorisometryMexuar roommotif+20 more

TL;DR

All 7 frieze symmetry groups (p111, p112, pm11, p1m1, pmm2, pmg2, and p1g1) are recoverable from a single building—the Alhambra in Granada, Spain—spanning materials as distinct as glazed tile, alicatado (cut tile), and yeso (plaster). Using the international four-symbol notation pxyz and the analytical method of identifying mirror reflections first, then rotational symmetry, then glide-reflections, each of the 7 groups is matched to a physical example at the palace. The p1g1 class, which requires glide-reflection symmetry along the translation axis but no rotation or mirror reflection, appears with notable rarity in planar Islamic mosaic tilings; corroborating evidence from Abas and Salman's statistical survey in Symmetries of Islamic Geometric Patterns shows that 2 of the 3 rarest wallpaper groups involve glide-reflections (pg and pmg), and a parallel survey of Seville's Real Alcázar found zero instances of p1g1. The pmg2 dado in Figure 6 is further distinguished by counterchange symmetry—background and foreground tiles sharing identical shapes. This implies that the rarity of glide-reflection symmetry is not accidental but reflects a systematic aesthetic or craft preference in the Islamic geometric tradition.

What to take away

  1. 1. All 7 mathematically possible frieze symmetry groups—p111, p112, pm11, p1m1, pmm2, pmg2, and p1g1—can each be identified with at least one physical example in the Alhambra, Granada, Spain.
  2. 2. The international four-symbol notation pxyz is the classification instrument used: the first symbol p denotes a primitive cell, the second encodes vertical mirror status (m or 1), the third encodes horizontal mirror or glide-reflection status (m, 1, or g), and the fourth encodes n-fold rotational symmetry (1 or 2).
  3. 3. The p1g1 frieze group, which has glide-reflection symmetry but no rotation or reflection, appears very rarely in planar Islamic mosaic tilings at the Alhambra and was entirely absent from a survey of Seville's Real Alcázar.
  4. 4. In Abas and Salman's Symmetries of Islamic Geometrical Patterns (World Scientific, 1998), 2 of the 3 rarest wallpaper symmetry groups involve glide-reflections (pg and pmg), supporting the hypothesis that glide-reflection is a systematically disfavored isometry in Islamic geometric art.
  5. 5. The Alhambra examples span 3 distinct substrate types—glazed tile, alicatado (cut tile derived from the Arabic qata'a), and yeso (plaster)—demonstrating that frieze symmetry classification is substrate-independent.
  6. 6. The analytical protocol introduced for classifying frieze patterns prioritizes identifying mirror reflections first, then rotational symmetry, then the subtler glide-reflection symmetry, a replicable sequential method applicable to any candidate one-dimensional periodic pattern.
  7. 7. The pmg2 example in Figure 6 (a glazed tile dado) exhibits counterchange symmetry, in which background and foreground tiles have identical shapes—a property notable enough to be explicitly flagged as distinct from ordinary pmg2 classification.
  8. 8. A p111 calligraphic inscription in the Mexuar room encodes the Nasrid motto 'There is no victor but Allah,' demonstrating that asymmetric (translation-only) frieze patterns at the Alhambra can carry explicit textual content rather than purely geometric motifs.
  9. 9. An open question the paper raises is whether the scarcity of glide-reflection friezes across multiple Iberian Islamic sites (Alhambra and Real Alcázar) reflects a broader pan-Islamic aesthetic preference or is a regional or period-specific phenomenon.
  10. 10. The seven frieze groups reduce to two-symbol shorthand notation (11, 12, m1, 1m, mm, mg, 1g) as a simplified alternative to the full four-symbol international symbols, a coding choice any researcher cataloguing pattern samples could adopt for efficiency.

Peer brief — for seminar discussion

The paper catalogues all 7 one-dimensional periodic frieze symmetry groups as instantiated in the decorative program of the Alhambra palace complex, Granada, Spain, using the international four-symbol pxyz notation as its classification instrument. Working across three substrate categories—glazed tile, alicatado (cut tile), and yeso (plaster)—the analysis matches each of the 7 groups (p111, p112, pm11, p1m1, pmm2, pmg2, p1g1) to at least one photographic example, identifies primitive cells, locates mirror axes, and marks 2-fold rotation centers. An alternative classification framework could have employed the two-symbol shorthand (11, 12, m1, 1m, mm, mg, 1g) throughout; the paper acknowledges both systems but conducts analysis in the full four-symbol form. The load-bearing finding is distributional rather than existential: while all 7 groups are present, the glide-reflection groups are rare. The p1g1 class is described as appearing very rarely in planar mosaic tilings at the Alhambra, and a prior survey of Seville's Real Alcázar (reported at the Joint Mathematics Meeting, January 2006, and at the MAA New Jersey Section, 2006) yielded zero p1g1 examples. This pattern is corroborated by Abas and Salman's statistical chart in Symmetries of Islamic Geometrical Patterns (World Scientific, 1998), which shows that 2 of the 3 rarest wallpaper groups involve glide-reflections (pg and pmg). The implied hypothesis is that glide-reflection symmetry is systematically disfavored in the Islamic geometric aesthetic, not merely absent by chance. An additional structural observation is that the pmg2 dado example exhibits counterchange symmetry—identical shapes for background and foreground tiles—a property the paper treats as aesthetically significant. A critical reader would push back on the evidentiary base: the claim that p1g1 is rare rests on informal observation ('this author was unable to find any examples') rather than a systematic exhaustive census of either building's ornamental program. Without a denominator—total number of frieze bands surveyed, proportion of each type identified—the distributional argument is anecdotal. The statistical support from Abas and Salman concerns wallpaper (2D) groups, and inferring that rarity of glide-reflections in 2D tilings predicts rarity in 1D friezes is an assumption, not a demonstrated result. The paper also does not engage with whether restored versus original panels were treated equivalently, which could confound any frequency claims.

Methods (1)

Frameworks (2)

  • 17 wallpaper groups
    Classification of two-dimensional repeating patterns into 17 symmetry groups
  • Seven frieze pattern groups
    Classification of one-dimensional periodic patterns into 7 symmetry groups using international notation pxyz

Findings (10)

Claims (5)

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  • alexander
    **Frieze Patterns of the Alhambra**papers/extracted/2022-06-06_Prabros._bridges2007-203.pdf_6211a5.md0.812