On Euler’s magic matrices of sizes 3 and 8

Summary. It is known that if M is a 3 × 3 magic matrix then every positive, odd power of M is a magic matrix, while an even power need not be. Given any n × n magic matrix M, we find all polynomials f such that f(M) is a magic matrix.

Complex and robust self-organization requires defined initial conditions and dynamic boundaries – neighboring tissues and extracellular matrix (ECM) that actively evolve to guide morphogenesis. A major challenge in tissue engineering is identifying material properties that mimic dynamic tissue boundaries but that are compatible with the engineering tools necessary for controlling the initial conditions of culture. Here we describe a highly tunable granular biomaterial, MAGIC matrix, that supports long-term bioprinting and gold-standard tissue self-organization. MAGIC matrix is designed for two temperature regimes: at 4 °C it exhibits reversible yield-stress behavior to support hours-long high-fidelity 3D printing without compromising cell viability; when transferred to cell culture at 37 °C, the material cross-links and exhibits viscoelasticity and stress relaxation that can be tuned to match numerous conditions, including that of reconstituted basement membrane matrices like Matrigel. We demonstrate that the timescale of stress relaxation and loss tangent are decoupled in MAGIC matrices, allowing us to test the role of stress relaxation rate and strain-dependence across formulations with identical storage and loss moduli. We find that fast absolute stress relaxation rates and large relative deformation magnitudes are required to optimize for morphogenesis. We apply optimized MAGIC matrices toward precise extrusion bioprinting of saturated cell suspensions directly into 3D culture. The ability to carefully control initial conditions for tissue growth yields dramatic increases in organoid reproducibility and complexity across multiple tissue types. We also fabricate perfusable 3D microphysiological systems that experience large strains in response to pressurization due to the compliant and dynamic tissue boundaries. Combined, our results both identify key parameters for optimal organoid morphogenesis in an engineered material and lay the foundation for fabricating more complex and reproducible tissue morphologies by canalizing their self-organization in both space and time.

At present, the Sudoku matrix, turtle shell matrix, and octagonal matrix have been put forward according to the magic matrix-based data hiding methods. Moreover, the magic matrices to be designed depend on the size of the embedding capacity. In addition, by determining the classification of points of pixel pairs after applying a magic matrix and by determining the traversal area, the average peak signal-to-noise ratio (PSNR) can be improved. Therefore, this topic intends to propose a data hiding method based on a 16 × 16 Sudoku matrix by using the 16 × 16 Sudoku matrix and extending it to a double-layer magic matrix. Low-cost data embedding methods are also studied, in order to improve the PSNR and maintain good image quality with the same embedding capacity.

We study a new class of matrices called diagonally magic matrices. We prove that such a matrix has rank at most 2 and that any square submatrix of a diagonally magic matrix is diagonally magic.