Mathematics Department Towson University Towson, MD 21252, USA
The Mathematical Institute Kneza Mihaila 35, 11001 Belgrade, Serbia & Montenegro
In In the present article, for our discussions, we mainly rely on two resources:
a treatise written by Buzjani in the 10 The treatise by Buzjani was originally written in Arabic, the academic
language of the Islamic world of the time, and was translated to the Persian
language of its time in two different periods: 10 Perhaps the most comprehensive and elegant recent book about Persian
mosaics that includes geometric constructions of designs presented along
with colorful images of their executions performed on different mediums
on the wall, floor, interior and exterior of domes, doors and windows,
and many more, is In the next section we introduce Buzjani by illustrating his "cut and
assemble" method of squares. In section 3 we present
what we propose as a modular art, based on the color contrast of sets of
cut-tiles. Section 4 proposes the possibility of "gaps"
and "overlaps" as two ideas for creating other ornamental designs based
on modularity. In section 5 we introduce Kufi modules
and show how this set has the capacity of generating a large number of
traditional ornamental and calligraphic designs.
Abul Wafa al-Buzjani was born in Buzjan, near Nishabur, a city in Khorasan,
Iran, in 940 A.D. He learned mathematics from his uncles and later on moved
to Baghdad when he was in his twenties. He flourished there as a great
mathematician and astronomer. He was given the title Buzjani’s important contributions include geometry and trigonometry. He was the first to show the generality of the sine theorem relative to spherical triangles and developed a new method of constructing sine tables. He introduced the secant and cosecant for the first time, knew the relations between the trigonometric lines, which are now used to define them, and undertook extensive studies on conics. In geometry he solved problems about compass and straightedge constructions in plane and in sphere. Buzjani wrote in In his treatise, in a chapter titled "
He continues:
Buzjani describes what he means by the mathematicians’ approach for solving this problem. In the following figure presents one side of a square unit. Then = , =, =, =, and so on. Therefore, in each step we are able to find the side of a square with its area equal to 2, 3, 4, 5, and so on. A square with side congruent to has the same area as 3 square units.
Then artisans presented several methods of cutting and assembling of these three squares. Some of these methods based on mathematical proofs turned out to be correct. Others were incorrect, even though they seemed correct at first glance. Buzjani illustrates two of these incorrect cutting-and-pasting constructions. We show one of them here: Some of the artisans locate one of these squares in the middle and divide the next one on its diagonal and divide the third square into one isosceles right triangle and two congruent trapezoids and assemble together as it seen in the figure.
For a layperson not familiar with the science of geometry, this solution seems correct. However, it can be shown that this is not the case. It is true that the resulting shape has four right angles. It is also true that each side of the larger shape seems to be one unit plus one half of the diagonal of the unit square. However, this construction does not result in a square because the diagonal of the unit square (which is the hypotenuse of the assembled larger triangle) is an irrational number but the measure of the line segment that this hypotenuse is located on in the larger shape is one and half a unit, which is a rational number. Buzjani includes more information and simple approximations to reinforce his point that the construction is not correct. However, his way of using an argument based on the idea of rational and irrational numbers is undoubtedly elegant. He then after presenting another incorrect artisans’ way of cutting and pasting gives his solution, which is mathematically correct. At the practical level it can be performed in any medium by artisans. But on the division of the squares based on reasoning we divide two squares along their diagonals. We locate each of these four triangles on one side of the third square such a way that one vertex of the acute angle of the triangle to be located on a vertex of the square. Then by means of line segments we join the vertices of the right angles of four triangles. From each larger triangle a smaller triangle will be cut using these line segments. We put each of these triangles in the congruent empty space next to it to complete the square.
Another interesting problem that Buzjani presents in his book is the composition of a single square from a finite number of different sizes of squares. For this, he solves the problem for two squares first and then comments that with the same method we are able to solve the problem for any number of squares.
The proposed solution for two squares is again based on cutting and pasting squares and therefore is acceptable by artisans. His solution is elegant: We first put the small square (a) on the top of the larger square (b) and then draw necessary line segments presented in (d) and (e) and finally cut the solid lines and paste them to obtain the resulting square.
For a person familiar with elementary mathematics what Buzjani is doing
in this problem can be justified as follows: Let
It may seem that some of the cutting and pasting activities presented
in So far we have seen that by using these modules one can approach ornamental qualities of patterns created by ingenious processes executed by either highly skilled artisans or mathematicians. What we wish to propose is controversial: There is the possibility that not very skilled artisans, who had no access to geometers, by the use of elementary cut-and-paste processes, based on trial and error, could create complex designs that require sophisticated geometric explanations—processes that remind us of modern fractal geometry: simple input with complicated results. The widespread approach for constructing and arranging pattern designs
in ceramic mosaic during the 10
One interesting Islamic pattern is the "maple leaf". Polya illustrated
the 17 wallpaper patterns in his article "Über die Analogie der Kristallsymmetrie
in der Ebene" published in Zeitschrift für Kristallographie in 1924.
He illustrated a maple leaf pattern and identified it as D
Figure 8 presents a traditional means of constructing the maple leaf pattern using compass and straightedge. In comparison we wish to present the modularity method and introduce the simplest possible set of modules using two single-color square tiles cut diagonally to generate the "maple leaf" pattern. If we cut a black and a white tile from their diagonals and exchange one of the generated triangles from each then we have a set of three modules of black, white, and half black-half white tile. In any other case by a diagonally shaped cut we create four modules as illustrated in Figure 6. Truchet’s 1704 paper laid down a mathematical framework for studying permutations based on these tiles [6].Now by using 14 white, 14 black, and 8 black-white tiles we create a 36-tile grid, which is a base for construction of the maple leaf tessellation (Figure 9).
Another interesting problem that Buzjani solved in assembling of squares
is if we have (
We should note that the "sum of two perfect square" problem, is in fact, a special case for the general case of "two different squares" problem, presented in Figure 4. Nevertheless the cut for the above two examples can provide us, or an artisan with a set of modules for exploring new patterns. The following figure presents a pattern, called "hat," which can be generated by only two opposite modules (without the use of original single color tiles) using the cuts from the above "5 squares" problem.
Figure 12 illustrates steps taken by a geometer or a highly skilled artisan to compose the "hat" grid using compass and straightedge. You may find more design constructions in [7].
The following tiling is generated from a set of modules that are cons-tructed from cutting and assembling thirteen squares as illustrated in Figure 10.
The treatise
In fact, tilings of two different size squares are not unusual in Islamic art. A proof of the Pythagorean theorem, attributed to Annairizi of Arabia (circa 900), is based on the tilling of two squares of different sizes (Figure 15). The bold overlay of larger squares presents copies of a square whose area is equal to the sum of the areas of the two square bases of the tiling. This proof is based on the cutting and assembling method [8].
Unlike "gapping," when we work with actual tiles, we cannot produce
"overlapping" during execution of an ornamental design on the wall. However,
one step earlier, before the artisan works with actual tile, he needs to
transfer the grid from his scrolls to the wall. In this step he can simply
overlay several grids to produce a more detailed and attractive tiling—a
perfect "overlapping." The artisans of medieval Persia were fascinated
to incorporate multiple-level designs into their ornaments. One to mention
among various methods was to use a smaller scale of the primary grid as
an overlay in a way that vertices of the two grids coincide. "An example
of an Islamic self-similar design is probably compiled in the late 15 The following figure presents the "maple leaf" tiling by means of a totally new construction. Here we have employed a different set of modules. Unlike the previous approach (Figure 9), we cannot create the design without using gaps and overlaps. It seems that these modules provide more capacity for creating new designs if we use proper gaps and overlaps. A person may discover this or similar tiling motif faster than using, for example, the modules that we introduced in Figure 10.
The modules that we introduced in Figure 16 are produced from a simple procedure of two equal cuts in two opposite color tiles and the exchange of opposite color triangles. We will use these modules not only to regenerate some original Persian ornamental designs, but also to render angular and geometric Kufic calligraphy of this area. Because of the relationship to Kufic calligraphy, we have chosen to call these Kufic modules.
Calligraphic design
using Kufi modules and a related tiling in the mausoleum of Shahzadeh Hossein,
Ghazvin, Iran.
Design using a combination
of overlapping Kufi modules and checkerboard tile,
and a similar tiling in the mausoleum of Shahzadeh Ibrahim, Isfahan,
Iran.
Design based on Kufi
modules and checkerboard tile, and a similar tiling in the Ali Mosque in
Esfahan, Iran.Using Kufi modules, appropriate gaps or overlaps, and appropriate use
of other tiles, we are able to generate the basic ornamental pattern for
a large class of tiling designs as we can observe in Figures
18-22. All the original patterns in these figures are from [3].
The art of Medieval Persian artists and artisans demonstrates the complex
overlay of geometric patterns, floral designs, and calligraphy. They achieved
this level of sophistication by collaboration with mathematicians of their
time and by improvement of their skill levels in geometric constructions.
The most ubiquitous method in creation of the ornamental patterns for the
artisans was by means of polygonal constructions. Some cut-tile patterns
suggest a modular approach, which is based on color contrast and repetition
by trial and error methods.
[1] A. Özdural, [2] S. A. Jazbi (translator and editor),
[3] M. Maher AnNaghsh, [4] R. Sarhangi, [5] B. D. Martin, R. Sarhangi, [6] D. E. Smith, The tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy, Leonardo 20, 4, 1987, pp. 373-385. [7] B. L. Bodner, [8] R. B. Nelson, [9] J. Bonner, |