# Bernat Espigulé

Bernat lives in Catalonia and works both as a tech consultant and a mathematical researcher at Universitat de Barcelona. This involves spreading his love of maths via research articles, animations, 3D-printed sculptures, workshops, and school visits. Currently, Bernat is interested in exploiting the notion of complex tree to tackle unsolved problems in the field of complex dynamics, and to help develop the theory of analysis on fractals. His research on the theory of complex trees is available at www.ComplexTrees.com And his 3D artwork created using this theory is available at www.shapeways.com/shops/complextrees .

This special quaternary tree was created back in 2012 using Mathematica and 3D-printed in 2013 by Shapeways. Its tipset has an integer fractal dimension, D=log(1/4)/log(1/2)=2, which can be appreciated by looking at the square tree T{i/2,1/2,-i/2,-1/2} casted by its shadow. Its creation was motivated in part by a Sierpinski Tetrahedron sculpture with metal spheres made by Susanne Krömker, head of Visualization and Numerical Geometry Group - IWR. The tetrahedral tree is a 3D version of the ternary tree with a Sierpinski triangle tipset, T{(-1+i/√3)/4,1/2,(-1-i/√3)/4}. This fact led Poland Embassies to wrongly attribute its inception to the Polish mathematician Waclaw Sierpinski, creating in 2015 several large-scaled copies of the tree.

This ternary complex tree T{i/√3,1/2-i/2√3,-1/2+i/2√3} represented in 3D as a three-legged table can be used to tile the plane. It was first created using Mathematica and 3D-printed by Shapeways back in 2013 as a gift to Robert Fathauer in relation to his "Three-Fold Development" ceramic sculpture, a fractal design that shares the same geometry. A variant of this tree is found in Benoit Mandelbrot's book "The Fractal Geometry of Nature" plate 73 as a plane-filling fractal tree called "fludgeflake". As a complex tree, see https://arxiv.org/abs/1902.11282 , it is structurally unstable, its reverse is a mirror-reflection of itself, and the fractal dimension of its tipset is D=log(1/3)/log(1/√3)=2.