Mozambique's Ethnomathematics Research Project (MERP), MOZAMBIQUE
The
rich cultural diversity of Africa is clearly visible in the wide range
of house decorations, of architectural styles and of settlement and
enclosure shapes. Unity within this diversity appears in the importance
of the artistic and geometrical exploration of symmetrical forms and
patterns. Shapes and decorations are not static: they may vary with the
seasons, mark changes in the family composition or be chosen for special
ceremonies. Some traditional African architectural ideas may have been
derived from or suggested by experience and knowledge in other cultural
spheres, such as basketry.Among the Ngongo, one of the ethnic-cultural groups of the Kuba Kingdom in central Congo/Zaire, the decoration of the walls of the houses and palaces with mat work was widespread. The plane patterns have various symmetries. Horizontally one sees the sticks which are woven together by the vertical lianas. The use of architectural mats is one way to change decorations in agreement with the season, ceremony or life cycle.
In Lesotho and neighboring zones of South Africa, Sotho women developed a tradition of decorating the walls of their houses with designs. The walls are fist neatly plastered with a mixture of mud and dung, and often colored with natural dyes. While the last coat of plastered mud is still wet, the women engrave the walls, using their forefingers. Their art is seasonal: the sun dries it and cracks it, and the rain washes it away. The entire village is redecorated before special religious celebrations such as engagement parties and weddings. The Sotho women call their geometric patterns litema. Symmetry is a basic feature of the litema patterns. They are normally built up from basic squares. The Sotho women lay out a network of squares and then they reproduce the basic design in each square. Of then the symmetries are two-color symmetries: horizontal and vertical reflections about the sides of the squares reverse the colors.
The correct citation for this paper is:
Paulus Gerder, "On Some Geometrical and Architectural Ideas from African Art and Craft", pp. 75-86 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
Michael J. Ostwald
School of Architecture and Built Environment
Faculty of Engineering and Built Environment
University of Newcastle
New South Wales, AUSTRALIA 2308
In
September 1995 the Australian architectural practice Ashton Raggatt
McDougall (ARM) invited the eminent mathematician Roger Penrose to open
their soon-to-be-completed refurbishment of the historic Storey Hall
complex of buildings at the Royal Melbourne Institute of Technology.
Penrose, who admitted that the design seemed "extremely exciting",
regretfully declined on the grounds that he was already overcommitted to
many projects to visit Australia at the required time. He concluded his
response to the invitation with an enigmatic postscript which records
that he is currently working on "the single tile problem" and recently
"found a tile set consisting of one tile together with complicated
matching rule that can be enforced with two small extra pieces". This
postscript contains the first clue to understanding the mysterious
connection between Penrose and Storey Hall, between a scientist and a
controversial, award-winning, building.
Storey Hall is significant for many reasons but only one prompted ARM to invite Penrose to open it. The newly completed Storey Hall is literally covered in a particular set of giant, aperiodic tiles that were discovered by Roger Penrose in the 1970's and have since become known as Penrose tiles. While architecture has, historically, always been closely associated with the crafts of tiling and patterning, Storey Hall represents a resurrection of that tradition.
But what is Penrose tiling and what does it have to do with architecture in general and Storey Hall in particular? This paper provides an overview of the special properties and characteristics of Penrose's tilings before describing the way in which they are used in ARM's Storey Hall. The purpose of this binary analysis is not to critique Storey Hall or its use of aperiodic tiling but to use ARM's design as a catalyst for taking the first few steps in a greater analysis of Penrose tiling in the context of architectural form generation.
ABOUT THE AUTHOR
Michael J. Ostwald is Dean of Architecture at the University of Newcastle, Australia. He has lectured in Asia, Europe and North America and has written and published extensively on the relationship between architecture, philosophy and geometry. He has a particular interest in fractal, topographic and computational geometry and has been awarded many international research grants in this field. Michael is also Co-Editor of the journal Architectural Design Research and was previously Book Review Editor of the Nexus Network Journal.
The correct citation for this paper is:
Michael J. Ostwald "Aperiodic Tiling, Penrose Tiling and the Generation of Architectural Forms", pp. 99-111 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
School of Architecture and Built Environment
Faculty of Engineering and Built Environment
University of Newcastle
New South Wales, AUSTRALIA 2308
In
September 1995 the Australian architectural practice Ashton Raggatt
McDougall (ARM) invited the eminent mathematician Roger Penrose to open
their soon-to-be-completed refurbishment of the historic Storey Hall
complex of buildings at the Royal Melbourne Institute of Technology.
Penrose, who admitted that the design seemed "extremely exciting",
regretfully declined on the grounds that he was already overcommitted to
many projects to visit Australia at the required time. He concluded his
response to the invitation with an enigmatic postscript which records
that he is currently working on "the single tile problem" and recently
"found a tile set consisting of one tile together with complicated
matching rule that can be enforced with two small extra pieces". This
postscript contains the first clue to understanding the mysterious
connection between Penrose and Storey Hall, between a scientist and a
controversial, award-winning, building.Storey Hall is significant for many reasons but only one prompted ARM to invite Penrose to open it. The newly completed Storey Hall is literally covered in a particular set of giant, aperiodic tiles that were discovered by Roger Penrose in the 1970's and have since become known as Penrose tiles. While architecture has, historically, always been closely associated with the crafts of tiling and patterning, Storey Hall represents a resurrection of that tradition.
But what is Penrose tiling and what does it have to do with architecture in general and Storey Hall in particular? This paper provides an overview of the special properties and characteristics of Penrose's tilings before describing the way in which they are used in ARM's Storey Hall. The purpose of this binary analysis is not to critique Storey Hall or its use of aperiodic tiling but to use ARM's design as a catalyst for taking the first few steps in a greater analysis of Penrose tiling in the context of architectural form generation.
ABOUT THE AUTHOR
Michael J. Ostwald is Dean of Architecture at the University of Newcastle, Australia. He has lectured in Asia, Europe and North America and has written and published extensively on the relationship between architecture, philosophy and geometry. He has a particular interest in fractal, topographic and computational geometry and has been awarded many international research grants in this field. Michael is also Co-Editor of the journal Architectural Design Research and was previously Book Review Editor of the Nexus Network Journal.
The correct citation for this paper is:
Michael J. Ostwald "Aperiodic Tiling, Penrose Tiling and the Generation of Architectural Forms", pp. 99-111 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
Gert Sperling
Lerchenweg 3
34233 Fuldatal, GERMANY
The
Pantheon complex has been the object of countless interpretations.
There is no certainty as to how and why it was created and what it is
meant to express, because there are no documents concerning the identity
of the architect, the exact dates of conception, its origin or its
function. Since ancient times we find vague references to its symbolic
function: according to Dio Cassius, it resembles the heavens. But the
cosmological interpretations do not take into consideration the real
metrical dimensions of the whole complex nor the relation between its
numbers, shapes, forms and proportions. Even the modules are identified
very differently, so that it is difficult to compare the various
analyses.
Some scholars take the Neopythagorean roots of the Pantheon seriously, interpreting the architecture as an integrated visualization of the Greek mathematically-conceptualized theory of the cosmos, which consisted of an amalgamation of cosmological, geodetical and anthropomorphical dimensions. To generate harmony, the laws of arithmetic, geometry, astronomy and musical proportions are fused. The pantheon can be considered an architectural image of the Pythagorean cosmos, a "living organism" with a mathematically-proportioning "soul" and unchanging, "eternal" consonant-symphonic ratios. It "resembles the heavens", but is a resemblance based on mathematical knowledge, a summary of the ancient quadrivium.
ABOUT THE AUTHOR
Presbyterian minister Gert Sperling has researched topics concerning the Pantheon for over thirty years. He is well known from his many reports on the subject. His most complete work is his book, Das Pantheon in Rom: Abbild und Mass des Kosmos (Munich/Neuried: ars una Verlag, 1999).
The correct citation for this paper is:
Gert Sperling, "The Quadrivium in the Pantheon of Rome", pp. 127-142 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
Lerchenweg 3
34233 Fuldatal, GERMANY
The
Pantheon complex has been the object of countless interpretations.
There is no certainty as to how and why it was created and what it is
meant to express, because there are no documents concerning the identity
of the architect, the exact dates of conception, its origin or its
function. Since ancient times we find vague references to its symbolic
function: according to Dio Cassius, it resembles the heavens. But the
cosmological interpretations do not take into consideration the real
metrical dimensions of the whole complex nor the relation between its
numbers, shapes, forms and proportions. Even the modules are identified
very differently, so that it is difficult to compare the various
analyses.Some scholars take the Neopythagorean roots of the Pantheon seriously, interpreting the architecture as an integrated visualization of the Greek mathematically-conceptualized theory of the cosmos, which consisted of an amalgamation of cosmological, geodetical and anthropomorphical dimensions. To generate harmony, the laws of arithmetic, geometry, astronomy and musical proportions are fused. The pantheon can be considered an architectural image of the Pythagorean cosmos, a "living organism" with a mathematically-proportioning "soul" and unchanging, "eternal" consonant-symphonic ratios. It "resembles the heavens", but is a resemblance based on mathematical knowledge, a summary of the ancient quadrivium.
ABOUT THE AUTHOR
Presbyterian minister Gert Sperling has researched topics concerning the Pantheon for over thirty years. He is well known from his many reports on the subject. His most complete work is his book, Das Pantheon in Rom: Abbild und Mass des Kosmos (Munich/Neuried: ars una Verlag, 1999).
The correct citation for this paper is:
Gert Sperling, "The Quadrivium in the Pantheon of Rome", pp. 127-142 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
Graziella Federici Vescovini
Università degli Studi di Firenze, ITALY
One
of the single most important ideas emerging from Alberti's concepts is
that of the relationship between the artist's ingenuity and his natural
and social surroundings, that is, the relationship between the world and
the artist's representation of it. It has been noted that this concept
resonates singularly with that of Nicholas of Cusa, with whom Alberti
shared mutual friendships. The idea of the creativity of the artist's
mind, including thoughts on his relationship to the world around him,
his capacity to harmoniously reconstruct in accordance with innate
proportions (the mind is the locus of proportion, Nicholas wrote) and
the beauty of discordant, contradictory Nature, is clearly developed in De Mente and other works anterior to 1450, when Alberti labored over De re aedificatoria. Nicholas' idea of the architectonic vis of the human mind finds a singular consonance with Alberti's vis compositionis,
according to which the artist imitates the divine ars in recomposing
the contradictions, irregularities and even monstrosities of the world.
Neither Nicholas nor Alberti presents the concept of the relationship
between the artist and the world around him as something tranquil and
objectively given, but rather as a continuing tension. It is a personal
conquest by the artist's ingenuity that corrects and upholds the mira vis creatrix
of nature. In part, the painter must draw out of nature the beauty that
certainly exists but is not always apparent, and in part he must be
capable of drawing it out of himself.
The correct citation for this paper is:
Graziella Federici Vescovini, "Nicholas of Cusa, Alberti and the Architectonics of the Mind", pp. 159-171 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
Università degli Studi di Firenze, ITALY
One
of the single most important ideas emerging from Alberti's concepts is
that of the relationship between the artist's ingenuity and his natural
and social surroundings, that is, the relationship between the world and
the artist's representation of it. It has been noted that this concept
resonates singularly with that of Nicholas of Cusa, with whom Alberti
shared mutual friendships. The idea of the creativity of the artist's
mind, including thoughts on his relationship to the world around him,
his capacity to harmoniously reconstruct in accordance with innate
proportions (the mind is the locus of proportion, Nicholas wrote) and
the beauty of discordant, contradictory Nature, is clearly developed in De Mente and other works anterior to 1450, when Alberti labored over De re aedificatoria. Nicholas' idea of the architectonic vis of the human mind finds a singular consonance with Alberti's vis compositionis,
according to which the artist imitates the divine ars in recomposing
the contradictions, irregularities and even monstrosities of the world.
Neither Nicholas nor Alberti presents the concept of the relationship
between the artist and the world around him as something tranquil and
objectively given, but rather as a continuing tension. It is a personal
conquest by the artist's ingenuity that corrects and upholds the mira vis creatrix
of nature. In part, the painter must draw out of nature the beauty that
certainly exists but is not always apparent, and in part he must be
capable of drawing it out of himself.The correct citation for this paper is:
Graziella Federici Vescovini, "Nicholas of Cusa, Alberti and the Architectonics of the Mind", pp. 159-171 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
Michele Emmer
All
of us feel we have an exclusive and privileged relationship with
Venice. We all feel that a particular bridge, a certain street, a hidden
corner of the city is only for us, that we have discovered it, that no
one else knows about it. Each of us has a special memory of the city on
the water. From 1976 to 1990 I made 18 films in a series on Art and
Mathematics. Several of them were made in Venice, at least in part.
Being films, they had a strong visual element and there I had to ask
myself the basic question: are there objects, places or works of art in
Venice that are of mathematical and of architectural interest? The
answer of course is yes, at two different levels. As the city-theater par excellence,
one has only to move around to discover that the architectural
structures--palaces, streets and squares--have geometrical and
mathematical shapes of some importance.
There are specific elements in Venice that are of special interest to the history of mathematics: polyhedra, symmetry, spirals and labyrinths. Add to this the fact that some of these works of scientific interest were carried out by great artists of the Renaissance and one realizes that it is not altogether far-fetched to think of Venice when dealing with the mathematics of art and architecture.
The correct citation for this paper is:
Michele Emmer, "La Venezia Perfetta: the Geometry of the City", pp. 39-50 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
All
of us feel we have an exclusive and privileged relationship with
Venice. We all feel that a particular bridge, a certain street, a hidden
corner of the city is only for us, that we have discovered it, that no
one else knows about it. Each of us has a special memory of the city on
the water. From 1976 to 1990 I made 18 films in a series on Art and
Mathematics. Several of them were made in Venice, at least in part.
Being films, they had a strong visual element and there I had to ask
myself the basic question: are there objects, places or works of art in
Venice that are of mathematical and of architectural interest? The
answer of course is yes, at two different levels. As the city-theater par excellence,
one has only to move around to discover that the architectural
structures--palaces, streets and squares--have geometrical and
mathematical shapes of some importance.There are specific elements in Venice that are of special interest to the history of mathematics: polyhedra, symmetry, spirals and labyrinths. Add to this the fact that some of these works of scientific interest were carried out by great artists of the Renaissance and one realizes that it is not altogether far-fetched to think of Venice when dealing with the mathematics of art and architecture.
The correct citation for this paper is:
Michele Emmer, "La Venezia Perfetta: the Geometry of the City", pp. 39-50 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
Marco Frascari
Virginia Tech, USA
Livio Volpi Ghirardini
Mantua, ITALY
Are
architectural proportions metric, numeric, geometric or golden? Which
ones among the many in a building are the markers that should be
considered reference points for the proportioning of its parts? A golden
or divine magnifying glass that distorts rather than clarifies has been
applied to everything in the name of aesthetic and mystical impulses. A
proportion called the Golden Mean has long been the only explanation for a successive melange of proportions in all the visual arts. This Golden Mean
(also called the Divine Proportion) has been found repeatedly in the
pictures of growth patterns embodied in natural events or in the
pictures of human products. Since the last century it has so fascinated
mathematicians and artists that is is proposed by many as the absolute
aesthetic value.
By tracing lines onto pictures, this ideal proportion has been found in man-made artifacts and used to mark human achievements. As the acme of his mystically scientific process, pictures of the Parthenon with Golden sections traced on them have been exhibited as demonstrations of the beauty of its man-made, but nature-inspired, rational design. This graphic notion of beauty is so alluring and pervasive that it has been acritically forced upon us as an aesthetic paradigm since grade school.
The German, Apollonian search within the combined sciences of mathematics, philosophy and archaeology lies at the root of the scientific proposal of the Golden Mean as a panacea for explaining the composition of parts and foretelling the aesthetic future of man-made designs. German philosopher Adolf Zeising has made the Golden Mean the only possible principle of a scientific aesthetic and used the Parthenon with the usual diagram traced on it to provide the necessary archaeological authority for his theory of the omnipresence of the aesthetic guarantor phi. In 1876, in a ponderous article published in memory of Zeising, mathematician Siegmund Gunter reviewed Zeising's scientific aesthetics in a critical manner, but even he admitted that the presence of phi in ancient architecture, and notably in the Parthenon, was clear evidence of its being the powerful quintessence of classical aesthetic values. Without any doubt Zeising and Gunter were very skillful at measuring pictures, but it is clear that neither of them had ever measured a building following to tectonic principles.
The correct citation for this paper is:
Marco Frascari and Livio Volpi Ghirardini, "Contra Divinam Proportionem", pp. 65-74 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
Virginia Tech, USA
Livio Volpi Ghirardini
Mantua, ITALY
Are
architectural proportions metric, numeric, geometric or golden? Which
ones among the many in a building are the markers that should be
considered reference points for the proportioning of its parts? A golden
or divine magnifying glass that distorts rather than clarifies has been
applied to everything in the name of aesthetic and mystical impulses. A
proportion called the Golden Mean has long been the only explanation for a successive melange of proportions in all the visual arts. This Golden Mean
(also called the Divine Proportion) has been found repeatedly in the
pictures of growth patterns embodied in natural events or in the
pictures of human products. Since the last century it has so fascinated
mathematicians and artists that is is proposed by many as the absolute
aesthetic value.By tracing lines onto pictures, this ideal proportion has been found in man-made artifacts and used to mark human achievements. As the acme of his mystically scientific process, pictures of the Parthenon with Golden sections traced on them have been exhibited as demonstrations of the beauty of its man-made, but nature-inspired, rational design. This graphic notion of beauty is so alluring and pervasive that it has been acritically forced upon us as an aesthetic paradigm since grade school.
The German, Apollonian search within the combined sciences of mathematics, philosophy and archaeology lies at the root of the scientific proposal of the Golden Mean as a panacea for explaining the composition of parts and foretelling the aesthetic future of man-made designs. German philosopher Adolf Zeising has made the Golden Mean the only possible principle of a scientific aesthetic and used the Parthenon with the usual diagram traced on it to provide the necessary archaeological authority for his theory of the omnipresence of the aesthetic guarantor phi. In 1876, in a ponderous article published in memory of Zeising, mathematician Siegmund Gunter reviewed Zeising's scientific aesthetics in a critical manner, but even he admitted that the presence of phi in ancient architecture, and notably in the Parthenon, was clear evidence of its being the powerful quintessence of classical aesthetic values. Without any doubt Zeising and Gunter were very skillful at measuring pictures, but it is clear that neither of them had ever measured a building following to tectonic principles.
The correct citation for this paper is:
Marco Frascari and Livio Volpi Ghirardini, "Contra Divinam Proportionem", pp. 65-74 in Nexus II: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1998.
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