Kragujevac J. Math. 28 - 1 Kragujevac J. Math. 28 (2005) 7-17.


The Leuven Café Erasmus model and the Bristol Café Wall illusion

Leopold Verstraelen

K.U. Leuven - K.U. Brussel Center PADGE,
(Pure and Applied Differential Geometry),
Departement Wiskunde, Celestijnenlaan 200B, B 3001 Leuven, Belgium

Abstract. Intimite relations between the functioning of the natural world in general and of the human intelligence in particular have resulted in some mathematical models which give reasonable formal descriptions of some fundamental objects and processes occuring in nature as observed by human beings. These models constitute the essence of our general scientific knowledge and are illustrations of Feynman's saying that ''Nature speaks to us in the language of mathematics''.

And, in the words of Chern: ''While algebra and analysis provide the foundations of mathematics, geometry is at the core''. Geometry is the field of mathematics whose main source of intuition is human visual perception. So, it seems appropriate that geometry would contribute somewhat to a better understanding of visual perception. Paraphrasing Feynman, what follows may illustrate that ''Nature likes to be looked at with geometer's eyes and brains''.

Basically, a visual observation amounts to the recording of light-energy (further on called ''luminosity''). In mathematical terms this is well described by a surface (further on called ''visual-stimulus-surface''). Based on this visual information, our visual system (in the way this has been developed in our ancestors and in ourselves via their and our wider contacts with the observed realities of the surrounding world, and which evolutions indeed also have had and have influence on this recording of light-energy itself) makes us aware of a corresponding image which is our actual registration of this visual observation. And this image can essentially only be determined by the geometrical properties of this surface. My purpose here is to present this natural, and therefore simple, geometrical model in some more detail and to discuss a bit its application to some so-called visual illusions.