Gallery

Deformation of an FCC packing (first row) into an HCP packing (last row) with 3 intermediate configurations. For each row, the first column shows a schematic sketch of the configuration with the corresponding shifting angle; the second column shows a projection from above; the third column shows the corresponding Dgm0(f4) (blue) with red curves indicating the continuous change in Dgm0(f4) when the configuration is continuously deformed into the next row’s configuration.

The rhomboid tiling of 5 points on the real line. For example, the upper left 2-dimensional rhomboid defined. by (Ø,{A,B},{C,D,E}) is the convex hull of the points yØ, yA, yB, and y{A,B}. The horizontal line at depth k intersects the tiling in a geometric realization of the order-k Delaunay mosaic of the 5 points.
A distorted upper left quadrant in the persistence diagram. The lock and unlock operations have the effect of advancing or delaying the birth, and the fill and unfill operations have the effect of advancing or delaying the death.

The density fingerprint map of the hexagonal lattice. Left: the k-fold covers for four different radii: t = 0.25, 0.55, 0.75, 1.00. Right: the graph of the first nine density functions above the densigram, in which the zeroth function can be seen upside-down and the remaining density functions are accumulated from left to right.

A spherical triangle with vertices ni, nj, nk and the cap with center z whose boundary is the unique circle that passes through the three vertices. The spherical triangle is decomposed into three spherical quadrangles by connecting z to the midpoints of the three sides, and into six cones by further connecting z to the three vertices.
Projecting the 4-cube along a diagonal gives the rhombic dodecahedron in ℜ3, which we see decomposed into four distorted 3-cubes. The projection of the upper 2-skeleton of the 4-cube consists of the vertices, edges, and light blue facets shared by the distorted 3-cubes. Its boundary is a graph in the boundary of the dodecahedron, which we highlight with dark blue edges.

Decomposition of the plane into “land” and “water”. All geometric structures are for the same value of t: (a) sub-, super-, and level sets of three continuous functions; (b) sub- and superlevel sets of the discrete functions on the Voronoi tessellation and the Delaunay mosaic; (c) channel divided by level set of piecewise linear function; (d) level sets of piecewise linear functions, with square boxes marking the neighborhoods of a non-critical point, a minimum, a saddle, and a maximum.

Distances along a line.
Distances in the plane.

The densities of distances between adjacent vertices in weighted Delaunay mosaics: the length of intervals along the line on the left and the length of edges of the mosaic in the plane on the right. The weighted tessellations are constructed as slices of unweighted Voronoi tessellations in n = 2, 3, 5, 8, 13, 21, 34 dimensions.