Automaticity and Invariant Measures of Linear Cellular Automata

We show that spacetime diagrams of linear cellular automata $\unicode[STIX]{x1D6F7}:\,\mathbb{F}_{p}^{\mathbb{Z}}\rightarrow \mathbb{F}_{p}^{\mathbb{Z}}$ with $(-p)$-automatic initial conditions are automatic. This extends existing results on initial conditions that are eventually constant. Each automatic spacetime diagram defines a $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant subset of $\mathbb{F}_{p}^{\mathbb{Z}}$, where $\unicode[STIX]{x1D70E}$ is the left shift map, and if the initial condition is not eventually periodic, then this invariant set is nontrivial. For the Ledrappier cellular automaton we construct a family of nontrivial $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant measures on $\mathbb{F}_{3}^{\mathbb{Z}}$. Finally, given a linear cellular automaton $\unicode[STIX]{x1D6F7}$, we construct a nontrivial $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant measure on $\mathbb{F}_{p}^{\mathbb{Z}}$ for all but finitely many $p$.

Neither past nor present : Authenticity and late twentieth-century architectural heritage

It was the present moment. No one need wonder that Orlando started, pressed her hand to her heart, and turned pale. For what more terrifying revelation can there be than that it is the present moment? That we survive the shock at all is only possible because the past shelters us on one side and the future on another. But we have no time now for reflections.(Virginia Woolf, Orlando)How long does the present moment last? Where and when does the past begin and how does the present end? In physics – or more precisely in the special theory of relativity – the present can be defined as the coordinate origin in a spacetime diagram – an unextended point that separates an observer’s past and future light cones. From that point of view, the present has no duration at all; the past instantly assimilates the future without any hesitation in between. However, time perception tells us that we actually experience a ‘here and now’. Psychologists believe that the time range we perceive as the present, the socalled specious present, lasts about three seconds – the interval duration after which the brain may be said to reset its attention. This is already infinitely more than no duration at all but this recognition is still not enough to explain concepts like the present time or ‘today’ as an indicator of the contemporary. In the domain of history, the present seems to be a much more complex construction. When we speak of phenomena as contemporary, we place them in an extended present. We concede that the present encompasses the recent past and the near future – a temporal range that provides a stage for the actions and reactions that shape our world.

The Twin Paradox

To prepare for the high‐speed world where Galilean relativity breaks down, we now practice the skill of thinking in different frames. Practicing this in our familiar lowspeed world will help us avoid cognitive overload when we enter the more counterintuitive high‐speed world. We examine two problems that illustrate the process of thinking in different frames. Te frst is a brain‐teaser that suddenly becomes easier in a particular frame; the second is a classic problem from introductory physics that we can solve in a new way using symmetry and frame-based thinking tools alone. Along the way, we will learn how to use the spacetime diagram, a major tool that will appear throughout this book. Te chapter concludes with another look at accelerated vs. inertial frames, from the spacetime diagram point of view.

Time Skew

This chapter explores one major consequence of the invariance of c: events that are simultaneous in one frame are not necessarily simultaneous in other frames.We will find that the time coordinates of events are just as frame‐dependent as their positions. This is no accident, but a symmetry between space and time. Viewed in a spacetime diagram, a frame change rotates the grid lines marking time just as much as it rotates the grid lines marking position; this preserves c as the same speed in all frames. Along the way, we practice using skewed grids in spacetime diagrams: identifying the time coordinates of events, identifying events that are simultaneous in a given frame, and adding velocities. Although the skewed grids change the time coordinates of events and even their order in time, we show that they do not change causal relationships between events.

Reasoning with Frames and Spacetime Diagrams

To prepare for the high‐speed world where Galilean relativity breaks down, we now practice the skill of thinking in different frames. Practicing this in our familiar lowspeed world will help us avoid cognitive overload when we enter the more counterintuitive high‐speed world. We examine two problems that illustrate the process of thinking in different frames. Te frst is a brain‐teaser that suddenly becomes easier in a particular frame; the second is a classic problem from introductory physics that we can solve in a new way using symmetry and frame‐based thinking tools alone. Along the way, we will learn how to use the spacetime diagram, a major tool that will appear throughout this book. Te chapter concludes with another look at accelerated vs. inertial frames, from the spacetime diagram point of view