Some of Dailey’s current research interests

Projects currently being focused on (spring 2018).  The first are the projects most active currently, though each of the first ten has received some sort of attention in the past five years. 
  1. Lexicographic adventures. See, for example, here, here, and here
  2. Prime oddities.  {3, 37, 373, 3733, 37337, 373357, 3733537, 33733537, 337335737, 3373357373, 33733537373, 337335337373} is a set of twelve primes that I call "orthographically descended" from one another, in the sense that the nth term is derived from the n-1st term by inserting (at any of the n positions available) a new digit. And they are all prime! This discussion provides an overview of definition and problems believed to be unsolved. For example, are there infinitely many such sequences?
  3. Visual paradox, tessellation, etc.

  4. Replicating in 3D

  5. Helping take grapher into the next phase of research on navigation (gravity in graphs) See gravity graphs below. This year the primary emphasis should be on visualization of "traffic" in graphs such as websites and social networks.

  6. Finding gravitational flavorings of graphs that motivate simplified navigation of those graphs. Humans use cues to navigate in our physical world. We use landmarks, compasses, and even our sense of gravity. Consider abstract metric spaces (like graphs). May we impose "field effects" like magnetism and gravity into graphs so that navigation is simplified? In particular might a single dimension of "flavoring" (gravity) of a graph , by assigning ordinal ranks to the nodes, allow a greedy algorithm to find a shortest path between any pair of nodes p and q in exactly d(p, q) steps? It turns out that the N-lattice is flavorable with a single dimension of gravity. See more at and

  7.  Build the Parisian street model of font-design and web layout.  See paper and software
    Consider using Voronoi tessellations, but basically, design a font-editor which also creates rich geometries for flowed content. Think: nonrectilinear layout models for HTML, CSS, and SVG. See also here for some allied issues. This year it would be nice to fix existing issues and to add the alternative of using Voronoi diagrams instead of line-arrangements.

  8. Web combinatorics: Rectangular and polygonal partitions of the plane. How many different HTML tables are there,  allowing for rotation and rescaling? How many different ways may a rectangle be carved into N rectangles? How many different ways may a rectangle be carved into N k-sided polygons? How might we, algorithmically, and quickly, choose a random partition of a rectangle into N k-sided polygons? One application: transitions (see below). Also issues like how to best to use SVG filters and their cousins in CSS to present things like Venn diagrams. (See also this.)

  9. Transitions: See these. Develop a large collection of open source transitions (fades, wipes, dissolves, etc.) from one image to another. Develop a simplified grammar to describe the class of all smooth inter-image transitions (including with memory for k previous images). Develop JavaScript which maps those transitions to  SVG, CSS3 and <canvas>.

  10. Extensions of the <replicate> proposal to SVG WG.
    See here for current progress). This year, we are contemplating an implementation that uses SVG Shadow DOM to prevent the actual DOM from being so terribly cluttered.

    Most active projects above.

    Less active projects below (though if someone is sufficiently interested, please hollder)
  11. working on string (and graph) similarity metrics w Dr. Whitfield and me. See any of us for a description, but you can glimpse some of it here:
    talk for PACISE

    The basic premise is this. Take a set of discrete structures (graphs, strings,  groups, discrete metric spaces). Consider the union of all their substructures. Measure the number of occurrences of each substructure within each structure. (For example, the number of occurances of K1 in a graph is its number of nodes; the occurances of K2 represents its number of lines.) Consider each substructure to be a dimension in a vector space. These measures of structures give to each structure a position in that vector space. The distances between any two structures can then be calculated using, for example, a Minkowski distance (Euclidean when r=2, Manhattan when r=1). If the Manhattan distance is used, then the resulting metric space is discrete, hence modelable by a graph. See embedding metric spaces in graphs, below.

  12. fast exponentiation of integers using Dailey numeric notation

  13. condensations of numeric fields of irrationals and

  14. psychophysical studies of equidistant objects

  15.  Each discrete metric space (in which all distances are integers) can have its metric properties exactly modeled by a graph, in the sense that graph theoretic distances (shortest paths) can perfectly represent the distances of the metric space (by introducing new "leprechaun" nodes -- sort of like inventing imaginary axes to describe certain simpler spaces, like Euclidean n-space). One question then is "what is the smallest graph that models a given metric space." In this 1994 paper, encouraged by and collaborated on with Frank Harary, I proved a few things including the very poetic sounding theorem: "the tree is the smallest container of the metric space defined on its leaves."  “On the Graphical Containment of Discrete Metric Spaces” David Dailey,.Discrete Mathematics, 1994, 131, 51-66. Elsevier Science.

  16. In a n-dimensional Euclidean space, construct a graph as follows: give each node a locus within that space, then connect any two nodes (i.e., draw a line between them) whose distance in that space is less than some threshold d.  Now what is the minimum dimensionality of the space in which a given graph can be realized subject to this constraint? Clearly Kn can be drawn in one dimension (just let d be large enough). Cn is two dimensional Wn the wheel on n nodes (a cycle of n-1 nodes each connected to a central spoke) becomes the problem of packing unit spheres. Frank Morgan of Williams College showed that any graph on n nodes is n-Euclidean. What else might we be able to say?

  17.  Client-side vectorization (using SVG filters). Maybe use <canvas> in conjunction with SVG? How to do efficient edge detection in images in the browser.

  18.  Organizing all the 1000 plus pages under the umbrella of
    The site has been growing for 15 years now (with much content predating that and having been created over 25 years at four institutions). It is all in need of reworking, reorganizing, styling and so forth.