The third (or organismic) view of space states that space is neither lifeless nor neutral, but a living structure capable of being more living or less living, thus different fundamentally from the first two mechanistic views of space: Newtonian absol
ute space and Leibnizian relational space. The living structure is defined as a physical and mathematical structure or simply characterized by the recurring notion (or inherent hierarchy) of far more small substructures than large ones. This paper seeks to lay out a new geography as a science of the Earths surface founded on the third view of space. The new geography aims not only to better understand geographic forms and processes but also - maybe more importantly - to make geographic space or the Earths surface to be living or more living. After introducing two fundamental laws of geography: Toblers law on spatial dependence (or homogeneity) and scaling law on spatial heterogeneity, we argue that these two laws are fundamental laws of living structure that favor statistics over exactitude, because the former (or statistics) tends to make a structure more living than the latter (or exactitute). We present the concept of living structure through some working examples and make it clear how a living structure differs from a non-living structure, under the organismic worldview that was first conceived by the British philosopher Alfred Whitehead (1861-1947). In order to make a structure or space living or more living, we introduce two design principles - differentiation and adaptation - using two paintings and two city plans. The new geography is a science of living structure, dealing with a wide range of scales, from the smallest scale of ornaments on walls to the scale of the entire Earths surface. Keywords: Scaling law, Toblers law, differentiation, adaptation, head/tail breaks, natural streets, the third view of space
When one agent interacts with a multi-agent environment, it is challenging to deal with various opponents unseen before. Modeling the behaviors, goals, or beliefs of opponents could help the agent adjust its policy to adapt to different opponents. In
addition, it is also important to consider opponents who are learning simultaneously or capable of reasoning. However, existing work usually tackles only one of the aforementioned types of opponent. In this paper, we propose model-based opponent modeling (MBOM), which employs the environment model to adapt to all kinds of opponent. MBOM simulates the recursive reasoning process in the environment model and imagines a set of improving opponent policies. To effectively and accurately represent the opponent policy, MBOM further mixes the imagined opponent policies according to the similarity with the real behaviors of opponents. Empirically, we show that MBOM achieves more effective adaptation than existing methods in competitive and cooperative environments, respectively with different types of opponent, i.e., fixed policy, naive learner, and reasoning learner.
In multi-agent reinforcement learning, the inherent non-stationarity of the environment caused by other agents actions posed significant difficulties for an agent to learn a good policy independently. One way to deal with non-stationarity is agent mo
deling, by which the agent takes into consideration the influence of other agents policies. Most existing work relies on predicting other agents actions or goals, or discriminating between their policies. However, such modeling fails to capture the similarities and differences between policies simultaneously and thus cannot provide useful information when generalizing to unseen policies. To address this, we propose a general method to learn representations of other agents policies via the joint-action distributions sampled in interactions. The similarities and differences between policies are naturally captured by the policy distance inferred from the joint-action distributions and deliberately reflected in the learned representations. Agents conditioned on the policy representations can well generalize to unseen agents. We empirically demonstrate that our method outperforms existing work in multi-agent tasks when facing unseen agents.
Twisted moire superlattices (TMSs) are fascinating materials with exotic physical properties. Despite tremendous studies on electronic, photonic and phononic TMSs, it has never been witnessed that TMSs can exhibit higher-order band topology. Here, we
report on the experimental observation of higher-order topological states in acoustic TMSs. By introducing moire twisting in bilayer honeycomb lattices of coupled acoustic resonators, we find a regime with designed interlayer couplings where a sizable band gap with higher-order topology emerges. This higher-order topological phase host unique topological edge and corner states, which can be understood via the Wannier centers of the acoustic Bloch bands below the band gap. We confirm experimentally the higher-order band topology by characterizing the edge and corner states using acoustic pump-probe measurements. With complementary theory and experiments, our study opens a pathway toward band topology in TMSs.
Gauge fields are at the heart of the fundamental science of our universe and various materials. For instance, Laughlins gedanken experiment of gauge flux insertion played a major role in understanding the quantum Hall effects. Gauge flux insertion in
to a single unit-cell, though crucial for detecting exotic quantum phases and for the ultimate control of quantum dynamics and classical waves, however, has not yet been achieved in laboratory. Here, we report on the experimental realization of gauge flux insertion into a single plaquette in a lattice system with the gauge phase ranging from 0 to 2pi which is realized through a novel approach based on three consecutive procedures: the dimension extension, creating an engineered dislocation and the dimensional reduction. Furthermore, we discover that the single-plaquette gauge flux insertion leads to a new phenomenon termed as the topological Wannier cycles, i.e., the cyclic spectral flows across multiple band gaps which are manifested as the topological boundary states (TBSs) on the plaquette. Such topological Wannier cycles emerge only if the Wannier centers are enclosed by the flux-carrying plaquette. Exploiting acoustic metamaterials and versatile pump-probe measurements, we observe the topological Wannier cycles by detecting the TBSs in various ways and confirm the single-plaquette gauge flux insertion by measuring the gauge phase accumulation on the plaquette. Our work unveils an unprecedented regime for lattice gauge systems and a fundamental topological response which could empower future studies on artificial gauge fields and topological materials.
Topological phases of matter lie at the heart of physics, connecting elegant mathematical principles to real materials that are believed to shape future electronic and quantum computing technologies. To date, studies in this discipline have almost ex
clusively been restricted to single-gap band topology because of the Fermi-Dirac filling effect. Here, we theoretically analyze and experimentally confirm a novel class of multi-gap topological phases, which we will refer to as non-Abelian topological semimetals, on kagome geometries. These unprecedented forms of matter depend on the notion of Euler class and frame charges which arise due to non-Abelian charge conversion processes when band nodes of different gaps are braided along each other in momentum space. We identify such exotic phenomena in acoustic metamaterials and uncover a rich topological phase diagram induced by the creation, braiding and recombination of band nodes. Using pump-probe measurements, we verify the non-Abelian charge conversion processes where topological charges of nodes are transferred from one gap to another. Moreover, in such processes, we discover symmetry-enforced intermediate phases featuring triply-degenerate band nodes with unique dispersions that are directly linked to the multi-gap topological invariants. Furthermore, we confirm that edge states can faithfully characterize the multi-gap topological phase diagram. Our study unveils a new regime of topological phases where multi-gap topology and non-Abelian charges of band nodes play a crucial role in understanding semimetals with inter-connected multiple bands.
To say that beauty is in the eye of the beholder means that beauty is largely subjective so varies from person to person. While the subjectivity view is commonly held, there is also an objectivity view that seeks to measure beauty or aesthetics in so
me quantitative manners. Christopher Alexander has long discovered that beauty or coherence highly correlates to the number of subsymmetries or substructures and demonstrated that there is a shared notion of beauty - structural beauty - among people and even different peoples, regardless of their faiths, cultures, and ethnicities. This notion of structural beauty arises directly out of living structure or wholeness, a physical and mathematical structure that underlies all space and matter. Based on the concept of living structure, this paper develops an approach for computing the structural beauty or life of an image (L) based on the number of automatically derived substructures (S) and their inherent hierarchy (H). To verify this approach, we conducted a series of case studies applied to eight pairs of images including Leonardo da Vincis Mona Lisa and Jackson Pollocks Blue Poles. We discovered among others that Blue Poles is more structurally beautiful than the Mona Lisa, and traditional buildings are in general more structurally beautiful than their modernist counterparts. This finding implies that goodness of things or images is largely a matter of fact rather than an opinion or personal preference as conventionally conceived. The research on structural beauty has deep implications on many disciplines, where beauty or aesthetics is a major concern such as image understanding and computer vision, architecture and urban design, humanities and arts, neurophysiology, and psychology. Keywords: Life; wholeness; figural goodness; head/tail breaks; computer vision
Sustainable urban design or planning is not a LEGO-like assembly of prefabricated elements, but an embryo-like growth with persistent differentiation and adaptation towards a coherent whole. The coherent whole has a striking character - called living
structure - that consists of far more small substructures than large ones. To detect the living structure, natural streets or axial lines have been previously adopted to be topologically represent an urban environment as a coherent whole. This paper develops a new approach to detecting the underlying living structure of urban environments. The approach takes an urban environment as a whole and recursively decomposes it into meaningful subwholes at different levels of hierarchy or scale ranging from the largest to the smallest. We compared the new approach to natural street and axial line approaches and demonstrated, through four case studies, that the new approach is better and more powerful. Based on the study, we further discuss how the new approach can be used not only for understanding, but also for effectively designing or planning the living structure of an urban environment to be more living or more livable. Keywords: Urban design or planning, structural beauty, space syntax, natural streets, life, wholeness
We demonstrate that multiple higher-order topological transitions can be triggered via the continuous change of the geometry in kagome photonic crystals composed of three dielectric rods. By tuning a single geometry parameter, the photonic corner and
edge states emerge or disappear with the higher-order topological transitions. Two distinct higher-order topological insulator phases and a normal insulator phase are revealed. Their topological indices are obtained from symmetry representations. A photonic analog of fractional corner charge is introduced to distinguish the two higher-order topological insulator phases. Our predictions can be readily realized and verified in configurable dielectric photonic crystals.
Photonic crystals have been demonstrated as a versatile platform for the study of topological phenomena. The recent discovery of higher order topological insulators introduces new aspects of topological photonic crystals which are yet to be explored.
Here, we propose a dielectric photonic crystal with unconventional higher order band topology. Besides the conventional spectral features of gapped edge states and in gap corner states, topological band theory predicts that the corner boundary of the higher-order topological insulator hosts a 2/3 fractional charge. We demonstrate that in the photonic crystal such a fractional charge can be verified from the local density of states of photons, through the concept of local spectral charge as an analog of the local electric charge due to band filling anomaly in electronic systems. Furthermore, we show that by introducing a disclination in the proposed photonic crystal, localized states and a 2/3 fractional spectral charge emerge around the disclination core, as the manifestation of the bulk disclination correspondence. The predicted effects can be readily observed in the state-of-the-art experiments and may lead to potential applications in integrated and quantum photonics.
