The recipe of a four-dimensional pastry: The hypercake

The first hyperchaotic attractor was discovered in 1979 by Otto E. Rössler. Since then, its structure has remained unexplored. Unveiling the topology of four-dimensional attractors is challenging because the tools commonly used in topology of chaos were developed mainly for chaotic attractors of flat structures, and most of the approaches do not apply in four dimensions. Since the Poincaré section of hyperchaotic attractors is not flat, and nor is their first return map, their structure cannot be reduced to a usual flat template. Moreover, chaotic attractors are organized around unstable periodic orbits, and for this reason, the knots of these orbits are commonly used to analyze, validate, and characterize the attractors structure. Unfortunately, knots unravel in the fourth dimension, rendering their use obsolete. In this study, the color tracer mapping is used to analyze the topology of the first hyperchaotic attractor. Its four-dimensional structure is disentangled, described, and validated. A subsampling-recomposing technique is then presented for transforming its four-dimensional structure into an edible cake. A detailed recipe is provided for its preparation in a traditional three-dimensional kitchen.