5th - 7th September, 2019 | Department of Mathematics of the Faculty of Sciences of Porto’s University, Portugal




Henry Segerman received his masters in mathematics from the University of Oxford, and his Ph.D. in mathematics from Stanford University. He is currently an associate professor in the Department of Mathematics at Oklahoma State University. His research interests are in three-dimensional geometry and topology, and in mathematical art and visualization. In visualization, he works mostly in the medium of 3D printing, with other interests in spherical video, virtual, and augmented reality. He is the author of the book "Visualizing Mathematics with 3D Printing".


I'll talk about my work in mathematical visualization: making accurate, effective, and beautiful pictures, models, and experiences of mathematical concepts. I'll discuss what it is that makes a visualization compelling, and show many examples in the medium of 3D printing, as well as some explorations in virtual reality and spherical video.


Born in Bilbao, where he studied and obtained a Ph.D. Degree in Computer Engineering at the University of Deusto.
Javier is Professor at the UPV-EHU (University of the Basque Country) where he teaches in the Department of Applied Mathematics at the School of Architecture in Donostia-San Sebastián.
Javier specialized in the relationship between Art & Science, dynamical systems including fractals and chaos, parametric programming of complex shapes and the scientific analysis for the structural maintenance and repair of heritage buildings.
Javier Barrallo has written over 100 papers and books, collaborated in 15 interventions on historical heritage buildings, organized 12 International Conferences and events and arranged 30 international exhibitions. 


Since ancient times polyhedra have exerted an enormous fascination not only in scientists but also in general population. They became the first three-dimensional geometric objects to be classified and, in our days, we are still surprised by their properties, aesthetics, variety and applications. Polyhedra are an educational element with enormous teaching possibilities and great attraction for students. It might seem that they do not have much theoretical importance in Mathematics or Geometry and that they lack utility and applications beyond its almost esoteric beauty, but this is not the case. From a theoretical point of view, polyhedra allow us to introduce basic terms such as vertex, edge, angle, polygon or symmetry, but also concepts such as convexity, duality, regularity, truncation and stellation, amongst others. Traditionally, the representation and construction of polyhedra was handcrafted but the popularization of CAD at the beginning of the 21st century and the emergence of 3D printers have left designing and manufacturing polyhedra in the hands of computers. Unfortunately, the mechanization in the assembly of polyhedra prevents taking advantage of many teaching experiences and skills present in the handcrafted creation of a polyhedron starting from scratch. Without wanting to despise all the innovation that has brought CAD / CAM and 3D printing, in this work we want to reclaim the artisanal construction of polyhedra with different materials, highlighting their interdisciplinarity by producing simple, enjoyable and accessible didactical projects for students of Mathematics, Fine Arts, Engineering, Architecture... as well as any other artistic or scientific discipline. We present polyhedral constructions involving topics like Fractals, Tensegrity, Fourth Dimension, Symmetry, Topology, Geodesics, etc. To demonstrate in a practical way the educational potential of polyhedra we present a collection of teaching projects carried out in the E.T.S. of Architecture of Donostia-San Sebastián at the University of the Basque Country.


Manuel Arala Chaves was Full Professor of Mathematics at the Faculty of Science of Porto from 1973 until 2003, when he decided to retire in order to work full-time in Atractor Association. Since the beginning, he has been the president of the board of this non-profit association, created in 1999 with the aim of raising public awareness and attracting to Mathematics. In 2000, Atractor was invited to create a large exhibition Matemática Viva. The exhibition lasted from November 2000 until August 2010 in Pavilhão do Conhecimento (Lisbon) and it had more than 2 million visitors. In more recent years, Atractor focused on producing interactive virtual contents and exhibits and many of them can be found on its homepage. Other activities include the DVD Symmetry the dynamical way (in 6 languages), several movies in Atractor's YouTube channel, a large collection of interactive stereoscopic 3D mathematical contents in several different formats (3D TVs, side-by-side, anaglyphs, etc.) and some free software, like GeCla and AtrMini (both in several languages).


The abundance and variety of azulejos in Portugal can be used in a form of cultural tourism which illustrates the mathematics of frieze and wallpaper patterns. As part of an ongoing project which we will describe, Atractor has developed tools and created a large database of existing azulejos, which will be easily accessible to visitors. It is well known that the number of types of symmetry for frieze and wallpaper patterns is not unlimited: in fact there are only 7+17 different possibilities. However, not even all of these are to be found in houses: there is a good reason for this, but it is not impossible to overcome, and it would be interesting if a town decided to have all possible patterns among its houses.
It is not obvious why there is a limit to the number of patterns, and we will give an intuitive idea of the reason. It is then natural to ask what happens in non-Euclidean geometries. We will see why the argument fails in this case and the result itself is not true.


Michael Hansmeyer is an architect and programmer who explores the use of algorithms to generate and fabricate architectural form. Recent work includes the design of two full-scale 3D printed sandstone grottos, the production of an iron and lace gazebo at the Gwangju Design Biennale, and the installation of a hall of columns at Grand Palais in Paris. He has exhibited at museums and venues including the Museum of Arts and Design New York, Palais de Tokyo, Martin Gropius Bau Berlin, and Design Miami / Basel. His work is part of the permanent collections of Centre Pompidou and FRAC Centre.
Recently, he taught architecture as visiting professor at the Academy of Fine Arts in Vienna and at Southeast University in Nanjing, and as a lecturer at the CAAD group of the Swiss Federal Institute of Technology (ETH) in Zurich. He previously worked for Herzog & de Meuron architects, as well as in the consulting and financial industries at McKinsey and J.P. Morgan respectively. Michael holds a Master of Architecture degree from Columbia University and an MBA from INSEAD.


Today, we can fabricate anything. Digital fabrication and 3D printing now function at both the micro and macro scales, and complexity and customization are no longer impediments in design. But can we fabricate more than we can design?
What is needed is a new type of design instrument. We need tools for search and exploration, rather than simply control and execution. They require a design language without the need for words and labels, as they should create the previously unseen. Knowledge and experience are acquired through search, demanding heuristics that work in the absence of categorization. These tools must ultimately redefine the process of design: the designer will work in an iterative feedback loop with the machine, moderating processes, and incorporating feedback, surprises and proposals. In this talk one such tool based on subdivision processes will be proposed.
As of yet, we have countless tools to increase our efficiency and precision. Why not also create tools that serve as our muse, that inspire us and help us to be creative? Tools to draw the undrawable, and to imagine the unimaginable.

RINUS ROELOFS (homepage)

Art about mathematics.
After studying mathematics for a couple of years (applied mathematics at the University of Twente, Enschede), I decided to switch to school of arts. And in 1983 I started my career as a sculptor. Inspired by the works of M.C. Escher and Leonardo da Vinci, my works became more and more some kind of expression of my mathematical ideas.
The main subject of my art is my fascination about mathematics. And to be more precise: my fascination about mathematical structures. Mathematical structures can be found all around us. We can see them everywhere in our daily live. The use of these structures as visual decoration is so common that we don’t even see this as mathematics. But studying the properties of these structures and especially the relation between the different structures can bring up questions. Questions that can be the start of interesting artistic explorations.
Artistic explorations of this kind mostly leads to intriguing designs of sculptural objects, which are then made in all kind of materials, like paper, wood, metal, acrylic, etc.. It all starts with amazement, trying to understand what you see. Solving those questions often leads to new ideas, new designs.
Since I use the computer as my main sketchbook these ideas come to reality first as a picture on the screen. From there I can decide what the next step towards physical realization has to be. A rendered picture, an animation or a 3D physical model made by the use of CNC-milling, laser cutting or rapid prototyping. Many techniques can be used nowadays, as well as many different materials. But it is all based on my fascination about mathematical structures.


The way most people know about the polyhedra is through physical experience (seeing, feeling) instead of theoretical understanding. To have a visual presentation of the polyhedra seems to be necessary for the understanding of the objects in this field of mathematics. Maybe more important than the real three-dimensional models are the two-dimensional pictures of these shapes. At some points in history artists added new ideas for presenting polyhedra in such a way that certain properties could be explained. Leonardo da Vinci made clear drawings of the elevated polyhedra as described by Luca Pacioli, Albrecht Dürer explained in his drawings how polyhedra can be made from 2D plans and M.C. Escher found nice ways to present the complex star-shaped polyhedra defined by Kepler. All these 2D representations were meant to show certain properties of the polyhedra.
In my presentation I want to show how the helix can be used to explain the complex uniform polyhedra with, in most cases, intersecting faces. Starting with the Poinsot polyhedra, which I will show with the use of method that I called “helixation”. This is a new dynamic way to visualize the structure of a polyhedron by showing how the polyhedron can be generated. With this technique we can also visualize things like duality.
In 2003 Branko Grünbaum described several ways to construct “new” uniform polyhedra. And again, there is a need for visualization of these new polyhedra. Also, in this case the helix turned out to be a good help to come to a clear visualization. In some cases, I used the shape of a torus knot, which is in fact a helix around a circle, to make the visualization.
Maybe the most important part of the talk is the use of the helix to construct a complete new group of uniform polyhedra, which I called the helical star deltahedra.

WORKSHOPS (to occur simultaneously)

WORKSHOP 1 - 07.09.2019    
This workshop relates closely with the above-mentioned lecture of the same title.It will consist in the assembly of some polyhedra featuring different materials and techniques. We pretend to recover didactical experiences with handcrafted polyhedra constructions that have been displaced in the last years by the explosion of three-dimensional printed models. Handcrafted polyhedra allow the introduction of mathematical and geometrical concepts in a very enjoyable and accessible way for the classroom. Both individual and group constructions during the workshop will feature connections with topics like Fractals, Topology or Tensegrity and will exercise the scientific and manual skills of the participants. No previous knowledge in the field is required, just the desire to play and experiment with geometry and take home beautiful polyhedra as souvenir.
WORKSHOP 2 - 07.09.2019    

Making paper models of regular polyhedra was, as far as I know, first described by Albrecht Dürer in his book “Unterweissung der Messung”, first published in 1525.
One of the pictures in this book is the folding plan of the icosahedron (Figure 1a). Most probably, Dürer was not aware of the fact that this folding plan also could be used to make another uniform polyhedron, the tetrahelix. One folding plan, two different polyhedra models!

A few simple exercises to continue with: two polyhedra combined in one model.
In the first one, we see the tetrahedron combined with a cube. The total model is made from 4 equal pieces. The second model is a so-called “double tetrahedron”, one model with a double skin. Also, the third model has a double skin: the “double-cube”. For both models we first need to weave the 2 basic parts     

Main course - first dish
In this part of the workshop, a new way of making double models is introduced. In the introduction, we started with just folding. In the starters, we have seen a combination of weaving and folding, and now the complete procedure is “folding-weaving-folding”. Again we will see combinations of 2 polyhedra in one model. The first model is the tetrahedron within the cube.
The second model is the cube within the dodecahedron. For this model two different versions can be made. The second one, with 3 equal parts has some connections with the Borremean Rings.
In the third picture, we see the models in which the outer skin is opened, in this way we can understand the total construction. In the workshop, both models will be available opened and closed.

Main course -second dish
We don’t have to limit ourselves to the Platonic solids. Kepler and Poinsot introduced new regular polyhedra which themselves have a double skin. Stellation is introduced by Kepler and in our model, we apply this technique onto the rhombic dodecahedron, a polyhedron with 12 faces. For our model the faces have to be extended to make the stellated version of the rhombic dodecahedron, which is also known as the Escher-star.

We will finish the workshop with a nice dessert. In 2003, Branko Grünbaum published some ideas with which new uniform can be created. One of his methods is called “doubling the faces”. When we double the faces of a cube, we can rearrange the connections of the squares in such a way that the 12 square faces will again form a regular polyhedron. The way of connecting the faces is in fact the same as our “double cube” model. In this model only some of the faces are double. The total model, again is a uniform polyhedron.

The drawing in the rightmost upper image is almost an exact representation of the great stellated dodecahedron inside a regular icosahedron.
Jamnitzer. W., 1568. Perspectiva Corporum Regularium

(optimized for googlechrome)