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Abstract The modern theory of quantized polarization has recently extended from 1D dipole moment to multipole moment, leading to the development from conventional topological insulators (TIs) to higher‐order TIs, i.e., from the bulk polarization as primary topological index, to the fractional corner charge as secondary topological index. The authors here extend this development by theoretically discovering a higher‐order end TI (HOETI) in a real projective lattice and experimentally verifying the prediction using topolectric circuits. A HOETI realizes a dipole‐symmetry‐protected phase in a higher‐dimensional space (conventionally in one dimension), which manifests as 0D topologically protected end states and a fractional end charge. The discovered bulk‐end correspondence reveals that the fractional end charge, which is proportional to the bulk topological invariant, can serve as a generic bulk probe of higher‐order topology. The authors identify the HOETI experimentally by the presence of localized end states and a fractional end charge. The results demonstrate the existence of fractional charges in non‐Euclidean manifolds and open new avenues for understanding the interplay between topological obstructions in real and momentum space.

Abstract Topological phases of matter are conventionally characterized by the bulk‐boundary correspondence in Hermitian systems. The topological invariant of the bulk in d dimensions corresponds to the number of (d − 1)‐dimensional boundary states. By extension, higher‐order topological insulators reveal a bulk‐edge‐corner correspondence, such that nth order topological phases feature (d − n)‐dimensional boundary states. The advent of non‐Hermitian topological systems sheds new light on the emergence of the non‐Hermitian skin effect (NHSE) with an extensive number of boundary modes under open boundary conditions. Still, the higher‐order NHSE remains largely unexplored, particularly in the experiment. An unsupervised approach—physics‐graph‐informed machine learning (PGIML)—to enhance the data mining ability of machine learning with limited domain knowledge is introduced. Through PGIML, the second‐order NHSE in a 2D non‐Hermitian topoelectrical circuit is experimentally demonstrated. The admittance spectra of the circuit exhibit an extensive number of corner skin modes and extreme sensitivity of the spectral flow to the boundary conditions. The violation of the conventional bulk‐boundary correspondence in the second‐order NHSE implies that modification of the topological band theory is inevitable in higher dimensional non‐Hermitian systems.

Injection molding is widely used in industries to produce polymeric products. At present, compound NURBS surfaces are commonly used to represent freeform surfaces in mold models. This work uses T-spline surface with extraordinary control points instead of NURBS to represent freeform surfaces in mold models. Compared with NURBS, T-splines’ higher-order continuity facilitates the mold quality control especially at the patch boundary. And, its patch layout information can be utilized for tool path planning. We propose an algorithm to determine the patch processing order and generate nonretraction tool path for T-spline surface models. The tool paths are generated patch by patch using isoparametric strategy. Actual machining and injection molding experiments have been conducted. The result shows the feasibility of the proposed method, and the final product is in good quality.