NumberProse Blog

Welcome back to my blog. In the next few blog posts, I will reflect on the mathematics of Architecture. This will also give me an opportunity to reminisce about travel adventures…

My husband's family gave us a trip to Barcelona for a special decade birthday present. One of the highlights of the trip was a visit to Gaudi's Sacrada Familia. Although, Gaudi never finished this impressive church, it is a good illustration of how his Architecture was influenced by mathematics and nature (Lorenzi and Francaviglia). 

Photo by Flickr/ Nhosko / Sagrada Familia

My first impression of Sacrada Familia was that the structure flows like fabric. I was impressed with how solid concrete appeared to be draping. Indeed, Burry writes that "Gaudi had two distinct strategies; sculpting and sewing". Burry was referring to some of Gaudi's design strategies. 

I was also drawn to the curvature in Gaudi's Architecture. Gaudi was interested in mathematics and how it could be applied to Architecture. Gaudi used his knowledge of mathematics, in particular geometrical curves and angles, and merged that with his observations of nature to create beautiful Art and Architecture. I recall from the museum in the Sacrada Familia, there is a model of construction that shows how Gaudi developed arches; he hung ropes from a ceiling and placed sacks of pellets on them. In this manner, the ropes and weights indicated the pressure points of the planned church arches. This U-shaped "chain curve" is referred to as the catenary curve.

Photo by Flickr/ Jon Olav / Catenary Arch

I think fondly back to my trip to Barcelona, for me the beautiful city was enhanced by Gaudi. 

Further Reading

There are many books and articles about Gaudi and his work, I have listed two here: 

J.burry and M.Burry, The New Mathematics of Architecture. Thames and Huddson, New York, 2010.

M.G. Lorenzi and M. Francaviglia, Art and Mathemathetics in Antoni Gaudi's Architecture:"La Sagrada Familia", Journal of Applied Mathematics, Volume 3, number 1

Photo by Flickr / yenidem

Beauty can be experienced in nature and the arts. What makes a sunflower or a pineapple visually appealing? Why do we feel that we are experiencing beauty when we walk up the spiral pathway in New York's Guggenheim Museum? Balance, harmony, symmetry, and proportion are often part of the interpretation of beauty. 

Frequently, the underlying proportion of beauty is the Golden Ratio. Simply defined, given a straight line, it is said to be cut in extreme and mean ratio such that the whole line to the greater segment is the same proportion as the greater line to the smaller line ( a+b:a is the same as a:b). This proportion is the Golden Ratio.

The Golden Ratio appears in nature, for example, in roses the positions of the petals are defined by fractional parts of multiples of the Golden Ratio. The nautilus shell spiral is a famous example of the Golden Ratio in the animal kingdom. The spiral of the nautilus shell is based on the Golden Ratio. In fact Frank Lloyd Wright uses the nautilus shell as his inspiration for the spiral ramp in the Guggenheim Museum in New York. 

The Golden Ratio is a fascinating number and has many mathematical properties. One of it's interesting properties is based on the Fibonacci Series;

1 1 2 3 5 8 13 21 34 55 89 144 233 …

Each term (starting with the third term) is equal to the sum of the preceding two terms. If you take the ratio of a term and the preceding term you get an approximation for the Golden Ratio. As you go further down the Fibonacci sequence the ratio of successive terms becomes closer and closer to the Golden Ratio. For a visual and audio treat watch this dreamlike animation which illustrates the Fibonacci Sequence and the Golden Ratio: Nature By Numbers (Cristobal Vila, 2010).

Livio (2002) points out that in nature the Golden Ratio may apply since it allows optimal spacing of plant growth.  The Golden Ratio is sometimes referred to as the divine proportion as some people associate it's appearance in nature with implications for the existence of God. Whether one considers the Golden Ratio to be a divine proportion or a practical ratio for optimal spacing in nature, it is undeniably an intriguing number which is often underlying in beauty.

Photo by Flickr / Guggenheim Museum Interior / loop_oh

Further Reading

In my earlier blog on Islamic Geometric Patterns, I describe how the Golden Ratio is present in pentagons which are frequently used in tiling.

For more in-depth examination of this fascinating ratio read Mario Livio's book, The Golden Ratio. 

Wolfram's A New Kind Of Science gives an alternative perspective on the arrangement of plant leaves (p408, Wolfram, 2002).

Flickr/Mandelbrot Set/Center For Image in Science and Art_UL

I often think of Britain fondly as it is my birthplace and I grew up there. It is a small country with a jagged coastline. Try going to google maps, and have a look at a map of Britain. You could approximate Britain's coastline length, by using a string to measure around the map of the coast, and then apply the scale factor. If you zoom in you will see a magnified view of Britain and with each successive zoom, you can see the coastline in greater detail. As you zoom in and measure the coastline with a string, the jagged nature of the coastline gives you a larger and larger approximation of the coastline. One could say that theoretically the distance around Britain is infinite! The late Benoit Mandelbrot wrote a paper titled How long is the Coast Line of Britian? Statistical Self-Similarity and Fractional Dimension (1967). In his paper Mandelbrot discusses the coastline paradox and his early thinking on fractals. "Fractals are defined as an object or quantity that displays self-similarity on all scales", (Wolfram).

Mandelbrot was a pioneer in recognizing that fractals appear in nature and there are practical applications of fractals. He created computer simulations of fractals and collaborated with people from many different fields on fractal applications. The famous Mandelbrot Set fractals were named after him. Although, the Mandelbrot Sets are based on a simple mathematical concept, their discovery was recent because computers are needed to perform the iterative mathematical computations to simulate the sets. Watch Arthur C. Clark's documentary, The Color of Infinity to learn more about Mandelbrot Sets, it is a musical and visual treat.

Infinity Sculpture by Jose de Rivera /Image from Flickr/Wallyg

A few years ago my husband and I spontaneously planned a trip to Boston during Fall. We drove from Washington to Boston and enjoyed the beautiful orange, red, and yellow fall foliage en route. Upon our arrival we found it difficult to find a vacant room in a hotel within a thirty-mile radius of the city center. At this point you may be wondering, is this a math-related topic? Well, if we had come across Hilbert's Hotel we would've been given a vacant room regardless of how many guests they had that evening.

Hilbert's Hotel is named after the famous mathematician David Hilbert (1862 - 1943) (Maor, The Story of Infinity, 1987). In Hilbert's Hotel, if all the rooms are occupied and a new guest arrives needing a room, the manager does not turn the guest away. He moves the person in room 1 to room 2, the person in room 2 to room 3, the person in room 3 to room 4 and so on. The new guest is given room 1. This hotel has an infinite number of rooms. This anecdote illustrates the concept of infinity. Although we have defined extremely large numbers such as ten raised to the power of 100 (the Googol), these large numbers are unrelated to Infinity. Infinity is not a number but a concept. As Eli Maor describes in his book The Story of Infinity, mathematicians have been fascinated by the paradoxes of Infinity. George Cantor (1845 - 1914)  was the first mathematician to formally acknowledge infinity and explored the concept using set theory.  Watch the BBC documentary The History of Mathematics 4 - To Infinity and Beyond for more details on the stories of Cantor and other famous mathematicians from the past. 

Many cultures use beautiful symmetrical patterns in their art. I find Indian Kolam patterns to be appealing due to their symmetry and curvature. Kolam are patterns that are created with rice flour (Ascher, 2002). Although the designs can be free form, typically geometric line drawings are created around a grid pattern of dots. Women in South India decorate the thresholds of their homes with Kolam designs; they are thought to bring prosperity to the home. In South India this design form is also known as Pulli Kolam or Neli Kolam . Pulli means dots and Neli means curvy. South Indians believe the dots are symbolical for the challenges of life and the curvy lines represent the journey (Siromoney, 1978). 

Islamic geometric patterns begin with a tessellation grid whereas Kolam designs start with a grid of dots. Symmetry is an important aspect of Kolam and Islamic patterns.

The mathematics in Kolam patterns includes recursion, and fractals (Ascher, 2002).

Wikimedia Commons/ Wilbert A. Minier

Welcome back to my blog.  Today I will continue on the theme of geometric patterns in different cultures. This time let’s go to Africa …

While studying aerial photographs of thatched roofs of homes in Tanzania Dr Ron Eglash (2005) realized that a geometric fractal pattern was used on the roofs. This inspired him to investigate further and research other areas of African culture where fractals were used.

Fractals were discovered in other areas of African culture such as traditional hair braiding and windscreen design. Listen to Dr Eglash’s entertaining and informative podcast for more details.  Read his book African Fractals which examines the contribution made by African culture to mathematics, architecture, design and even computer simulations. 

If you are a math teacher, you may be interested in African Fractals Teaching Materials.

I hope you are enjoying the math bytes! If you would like to comment on my blog, please use the feedback button.

Origami is a Japanese word that means paper folding.  It is an art form based on folding a square piece of paper to create three-dimensional structures. Last time, I talked about Islamic art, which is based on a circle, and geometric constructions using a ruler and compass are used to create patterns. In contrast Origami is based on a square and geometric creases are used to create a figure. 

Using creases and folds, angles are bisected and trisected. Mathematics is involved in Origami. Robert Lang discusses in his podcast, how Origami has evolved and how sophisticated mathematics is applied to the ancient art form.  

When I was around eight years old, I loved Origami. One day my son became interested in my Origami book and now he has a passion for it. Here’s a photo of one of his creations.

Mathematics is present in the art and architecture of many cultures. Revealing the underlying mathematics increases the appreciation of the art form, and the sophistication of the culture. Recently I wrote a grad paper focusing on geometric art in different cultures. The paper is my inspiration for the next few blog entries. 

Geometric patterns are used in decorative tiling and in architectural structures throughout the Muslim world. 

Morocco Pavilion Tile Work. Wikimedia Commons/Notjake 13

Eric Broug’s book, Islamic Geometric Patterns is full of beautiful patterns from around the middle east. He explains that the circle is the basis of many Islamic patterns. At first glance it may not be apparent that the starting point of the pattern is the circle. Regular polygons are inscribed in circles. Triangles, squares, hexagons, pentagons, and stars frequently feature in Islamic geometric patterns. 

Islamic craftsmen divided their surface into a grid of the same regular polygons and repeated motifs in the grid. The artists were using tessellations. A tessellation is the covering of a plane with identical shapes with no overlaps or gaps (Alexandre, 1994). Note, this creates a restriction on which polygons can be used since in order to fill the plane at each vertex the interior angle of the polygon must divide exactly into 360 degrees. The formula for calculating the interior angle of a regular polygon is 

where n is the number of sides of the polygon. Triangles, squares and hexagons can be used for a tessellation grid.It is impossible to tessellate a pentagon since the measure of the interior angle is 108 degrees and this does not divide exactly into 360 degrees. Although pentagons cannot be tessellated, they are used in geometric patterns by combining them with more than one shape in a semi-regular tessellation grid (Broug, 2008). 

In Critchlow’s book Islamic Patterns: An Analytical and Cosmological Approach, he discusses the harmonic proportions in Islamic art. In Islamic patterns, a square within a square is sometimes used, Critchlow highlights that the ratio of the edge of the inscribed square to the edge of the outside square is 1: square root of 2. Pentagons containing five pointed stars feature in Islamic geometric designs. The edge length of the pentagon to the arm of the star is the well known beautiful proportion known as the golden ratio;

Topkapi Palace, Turkey/ Wikimedia Commons/G. Dallorto

Hello!

From time to time I may chat about mathematics books that are aimed at the general public. I'll begin with a great title Is God a Mathematician? by Mario Livio. Dr Livio has written other popular mathematics books, and has a pleasing storytelling style.

Livio discusses the mystery of mathematics. How does mathematics describe so many aspects of our physical world? This leads to the profound question, is mathematics discovered or invented? Livio ponders this question by discussing mathematical ideas of philosophers, mathematicians, and scientists throughout history to present day. He gives us a brief selective history of the evolution of mathematics as we know it.

Livio refers to great mathematicians as "magicians". These great magicians are able to conjure up mathematics that describe the physical world. Sometimes they are deliberately trying to derive mathematical laws to explain nature and other times it is accidental. Livio focuses on four great mathematicians, Archimedes, Galileo, Descartes, and Newton. Having provided a historical perspective, Livio gives his opinon on the "unreasonable effectiveness of mathematics" in describing the natural world.

At this point I cannot say anymore! Of course, I don't want to give away the conclusion that Mario Livio comes to, that would be similar to a movie reviewer spoiling the film by revealing the ending. Read the book and see if you agree ...