Cosmatesque Mosaic and Conformal Mapping

Year: 2023 Authors: Steve Pomerantz

Core claim

Traditional and modern Cosmatesque mosaic designs can be modeled through conformal maps of planar tilings.

Topics

Cosmatesque ornament, conformal mapping, mosaic patterns, planar tilings, watercolor design

Domains

complex analysis, conformal maps, Möbius transformation, planar geometry, mosaic, ornament, watercolor painting, architectural pattern

Methods

exponential map, Möbius transformation, power map, tiling transformation

Media

digital pattern diagrams, mosaic renderings, watercolor paintings

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2023 Conference Proceedings

Cosmatesque Mosaic and Conformal Mapping

Steve Pomerantz

Saint Joseph University; spomerantz@sjny.edu, circleofsteve@gmail.com

Abstract

This article highlights an application of conformal mapping as a means of generating traditional and modern Cosmatesque patterns. Many traditional Cosmatesque pattern can be modelled using a conformal exponential map of one complex variable. Alternative conformal maps lead to more modern versions.

Cosmatesque, or Cosmati, refer to mosaic patterns that were prevalent in $12^{\text{th}}$ century Rome, that appear throughout Italy [2] and the United Kingdom [3]. The name originates from several generations of craftsmen in the Cosmati family. While this style of ornament is almost a thousand years old, craftsmen today still manufacture and restore these designs. These patterns appear mainly on church floors. The most characteristic patterns are referred to as guilloche, a serpentine arrangement of alternating braids and the quincunx, an arrangement of four circles around a central circle again connected by a braid-like design as illustrated in Figure 1.

(a)

(b) Figure 1: Typical (a) Guilloche and (b) Quincunx patterns from Capella Palatina in Palermo.

Within the braids, as pictured above, is a tiling pattern often recognized as a regular or uniform tiling, albeit transformed to fit inside an annular region. Many of these transformed tilings can be constructed as the result of a conformal mapping of a corresponding planar uniform tiling. Discussed below will be a particular tiling of triangles, hexagons and parallelograms.

In particular the types of patterns demonstrated below are based on a common underlying grid. On the left in Figure 2 is illustrated a design within a rectangle of arbitrary width $w$ and height $2\pi$ that is uniquely determined by several factors, namely $w_{\min }$ , $w_{\max }$ , arbitrarily chosen to differ by $w$ , an angle $\alpha$ , which measures the slope of the diagonal lines and an integer $n$ that determines the number of vertical partitions.

Pomerantz

As $\alpha$ and $n$ vary, the shape and complexity of the annular patterns will vary. While the graph on the left illustrates a collection of diagonal and vertical lines, the alternative shown in Figure 2b selectively omits every third diagonal and vertical line and thus reveals a tiling of parallelograms, triangles and hexagons.

(a)

(b) Figure 2: Tessellation of a strip before transformation with (a) all lines and (b) selective removals.

The conformal mapping $(x,y)\to ({\mathrm{e}}^x\cos (y),{\mathrm{e}}^x\sin (y))$ transforms the grid from the rectangular domain to the annular range [1]. A further constraint on the design will force the image of $y=0$ to be the same as that of $y=2\pi$ . Applying the exponential conformal map to the grids above produces the following patterns which are illustrated below in Figure 3 and 4 in graphic and mosaic form.

Figure 3: Conformal result of Figure 2(a).

Figure 4: Conformal result of Figure 2(b).

Cosmatesque Mosaic and Conformal Mapping

More complicated designs can be constructed based on alternative planar tilings that include octagons and circular shapes, all subject to the exponential map.

(a)

(b)

(c) Figure 5: A Collection of watercolor paintings by the author inspired by (a) the Westminster Abbey in London and (b) the Sistine Chapel in Rome. A collection of designs by the author is in (c).

As a modern alternative, other conformal maps of $z=x+\mathrm{i}y$ such as $z^{\mathrm{a}}$ or $(\mathrm{a}+\mathrm{b}z)/(\mathrm{c}+\mathrm{d}z)$ produce patterns as illustrated below in Figure 6 followed by the authors watercolor rendition.

Figure 6a: Transformation of Figure 2b by the Möbius transformation $(1-z)/(1+z)$

Pomerantz

Figure 6b: Transformation of Figure 2b by $z^3$

Figure 6c: Transformation of Figure 2b by $z^2$

Realizing these patterns as physical mosaics would be more complex owing to the absence of identical tile pieces but a computer-driven process could overcome this difficulty.

References

[1] T. Needham, “Visual Complex Analysis”, Oxford Press, 1997 [2] P. Pajres-Ayuela, “Cosmatesque Ornament: Flat Polychrome Geometric Patterns in Architecture”, W.W. Norton & Co. 2001 [3] W. Rodwell, D. Neal, “The Cosmatesque Mosaics of Westminster Abbey”, Oxbow Books, 2019.