Indefinite Inner Products and Elements of Formline Art

Ben Blohowiak

Updated: 2025-02-09

My recent computational exploration of abstract mathematical spaces produced graphs of functions that seem to resemble "formline" motifs.
I emphasize that such resemblance surprised me. As I wrote code to visualize recursive functions generalized to spaces beyond the complex plane, I did not know how outputs would look.
I have since developed conjectures regarding how certain formline-like elements may correspond to certain mathematical conditions; I have included some examples and exploratory findings below.

Although I use the term "formline" as coined by Bill Holm, as I understand it, such terminology may be augmented or eclipsed by taxonomic and language-preserving efforts on various fronts (e.g., the the Haida-language design vocabulary Hluugiitgaa Gwaai and Gandll Gyaagan Jaalen Edenshaw are developing around the Sgaajuu, the efforts of Xʼunei Lance Twitchell, etc.).

I suspected that there might be some connection between indefinite inner products and appearance of formline elements upon seeing graphs/visualizations of escape-time fractals (e.g., Mandelbrot, tricorn) generalized to the complex hexapolar plane (an abstract mathematical space).
Although the "natural" modulus of a nonzero complex hexapolar may be positive, zero, or nonpositive (AKA "indefinite"), it is also possible to compute a positive-definite magnitude from its components as per a generalized Euclidean approach. The "natural" indefinite modulus of the complex hexapolars yeilded formline-like elements in the examples below whereas Euclidean-like computations [not shown] did not. (My assessment is admittedly quite informal.)
Mandelbrot, Complex Hexapolar Tricorn, Complex Hexapolar
PNG of s2s2 Mandelbrot variation, exponent 2 PNG of s8s8 Tricorn variation, exponent 2
PNG of s8s8 Mandelbrot variation, exponent 2 PNG of s8s6 Tricorn variation, exponent 2

To ascertain what influence may be attributed to a "natural" indefinite modulus, I examined instances of Mandelbrot and tricorn functions in what may be the prototypical example of a planar geometric interpretation of a number system with an indefinite modulus, the many-named Minkowski/hyperbolic/split-complex plane.
Whereas Minkowski spacetime--yet another abstract space with an indefinite inner product--is typically represented by a total of four basis vectors, the split-complex plane may be thought of as a simplified version of Minkowski spacetime represented by two basis vectors.

With some notable exceptions, Mandelbrot functions generalized to the split-complex plane are infrequently discussed in the literature, perhaps because the Mandelbrot set so generalized is apparently lacking in the self-similarity at various scales otherwise characteristic of its counterparts in other abstract spaces (see Hayes or Blankers et al.).
It is worth noting that the split-Mandelbrot set per se is rectilinear, though it is also worth noting that in low iteration counts of otherwise fractal recursive functions, regions of the split-complex plane may seem to approximate rounded-corner, ovoid, or split-U/trigon elements. Observing such elements accruing through iterations can illustrate how certain shape outlines may be defined by cumulative aggregations of smaller parts.
Mandelbrot
(2nd power)
Mandelbrot
(4th power)
Tricorn
(2nd power)
Zoomed In this space intentionally left blank
Centered at Origin
In my review, I did not recognize analogs to finelines in the split-complex plane as readily as I did in the complex hexapolar plane. Yes, parallel line segments of varying width can be found (as is visible toward the end of this video). Curvilinear fineline-like elements in recursive/fractal functions seem like they might be specific to spaces other than the split-complex plane such as the complex hexapolar plane in which I initially observed them; in the absence of further argument or evidence, there is no reason to assume that such elements must be specific to multipolar spaces (i.e., spaces in which basis vectors may have more than two inverse directions of extension).

If you observe curvilinear fineline-like elements associated with mathematical functions, please contribute to the literature and/or let me know.

Questions? Feedback?
ben at ben blohowiak dot com