Human–Machine Interfaces (HMIs) in traditional automobiles are essential in connecting drivers, passengers, and vehicle systems. In automated vehicles, the HMI has become a critical component. A well-designed HMI facilitates effective human oversight, enhances situational awareness, and mitigates risks associated with system failures or unexpected scenarios. Simultaneously, it serves as a crucial safeguard against cyber threats, preventing unauthorized access and ensuring the integrity of vehicular operations in increasingly connected environments. This narrative review delves into the evolving landscape of automotive HMI design, emphasizing its role in enhancing user experience (UX) and safety. By exploring usability challenges, technological advancements, and the integration of rapidly evolving technologies such as AI (Artificial Intelligence), AR (Augmented Reality), and gesture-based controls, this study highlights how effective HMIs minimize cognitive load while maintaining functionality. Significant attention is given to the new challenges that arise from technological advancements in terms of security and safety.

The Collatz conjecture is a famous unsolved problem in mathematics, known for its deceptively simple rules that generate complex, unpredictable behaviour. It can be efficiently modelled using a Petri net that represents its inverse graph, where each place corresponds to an integer and each transition encodes an inverse rule. The net, constructed up to a bound n, reveals the tree-like structure of predecessors and highlights properties such as recurrence, reachability, and liveness. Token flows simulate possible trajectories towards 1. This formal approach enables the investigation of the problem through discrete event systems theory and opens perspectives for parametric or inductive extensions beyond the bounded domain. The model proposed provides a structured framework for visualising and analysing the inverse dynamics of the conjecture. Some key numerical results highlight the challenges of working within a finite domain: for $n_{max}=1000$, the constructed Petri net comprises 1000 places and 667 transitions, including 417 source nodes (no predecessors), 333 sink nodes (no successors), and 218 isolated orphans, i.e., nodes only reachable via $\mathrm{Div}2$ transitions with no incoming $3n+1$ edge.

Pascal’s triangle is a classical mathematical construct, historically studied for centuries, that organises binomial coefficients in a triangular array and serves as a cornerstone in combinatorics, algebra, and number theory. Herein, we propose to model it with Petri nets, a formal specification technique derived from discrete event systems. A minimal Petri net is created that generates Pascal’s triangle under a simple arithmetic rule. Token counts in each place coincide with binomial coefficients, providing a direct combinatorial interpretation. Two other classical structures emerge from this model: by colouring tokens depending on their parity, the Sierpiński triangle appears; by routing tokens randomly at each branching, the binomial distribution arises, converging to a Gaussian limit as depth increases. As a result, a single Petri construction unifies three mathematical objects: Pascal’s Triangle, Sierpiński’s Triangle, and the Gaussian distribution. This connection illustrates the invaluable potential of Petri nets as unifying tools for modelling discrete mathematical structures and beyond.

This work does not aim to solve the Collatz conjecture, but to reformulate it within a new analytical framework based on Jacobsthal sequences. This reformulation reveals structural regularities and symmetries that remain hidden in the traditional approach, offering a new mathematical perspective from which the Collatz-type problems may be further explored. This approach can be reproduced in System Engineering analyses, where alternative representation are often valuable for exposing hidden structure and improving understanding. Beyond its mathematical significance, the proposed framework may also offer conceptual and methodological benefits for the field of Information Technology (IT). Many problems in computer science involve the analysis of iterative processes, recursive algorithms, and discrete state transitions. The Collatz transformation itself can be viewed as a simple computational process governed by conditional rules and repeated iteration, which makes it an interesting model for studying algorithmic behavior. Sequence-based representations are widely used in areas such as data compression, cryptography, and pseudo-random number generation. The identification of structured recurrence relations and parity patterns within integer sequences can provide insights into the design of efficient algorithms for generating deterministic yet complex numerical sequences. Such sequences are frequently employed in cryptographic systems, simulation models, and distributed computing applications.

Mathematics, as Bertrand Russell noted, possesses both truth and beauty. In this work, we revisit the classical Fibonacci recurrence thanks to a minimal Petri net. Starting from a minimal layered construction that mirrors the second-order additive rule $F_n=F_{n-1}+F_{n-2}$, we show that the marking dynamics of the associated net generate a combinatorial triangle whose parity structure reveals a self-similar, Sierpiński-like pattern. To the best of our knowledge, this oblique fractal geometry has never been formally documented. We provide a formal definition of the underlying Petri net, analyse its computational properties, and explore the emergence of higher-order harmonics when token markings are considered modulo primes. The study highlights how a classical recurrence gives rise to previously unnoticed geometric regularities at the intersection of mathematics and art. Beyond its mathematical interest, the construction illustrates how minimal Petri net dynamics can be used as formal specification patterns for distributed, event-driven systems.