Lukas' Blog
lukas (at)

2017-05-17 ↗

It's been a busy month. My time in LA has been defined by job churn and trying on a lot of different roles, only to find that the things I'm really committed to—the things I want to improve on—are old ones. Being a mathematician, asking effective questions ("strategy"), fabrication & construction, interdisciplinary research/writing.

Hard to find entry-level work doing that, still, but I have feelers out and am also looking into graduate programs. If I can get by until a grad program, I think it could give me the momentum necessary to get to the next phase. Some people have shown me statistics/data-related jobs, which are interesting, but statistics is not mathematics. I get the impression many companies which would need math modeling work are filling that void with machine learning. So I suppose I am one of those people whose work is being automated. (For better or for worse? I'll keep that to myself...)

My only two commitments right now are lots of math tutoring and a slew of design & research for The Cactus Store. I helped make the Super Kabuto site and am learning more about these extraordinary plants.

Math tutoring has been the only constant in LA—I've been tutoring the same students, through the same company, since September. I love teaching math, and have had a strong interest in its intricacies since reading A Mathematician's Lament in high school.

Teaching any science is mostly about helping a student build an effective mental model of the mechanics at play. (The line between "mental model" and "metaphor" is thin—I think the former needs to keep a close structural similarity, whereas the latter has more room for suggestive analogy.) And math is only a science in as much as it can describe the world; its depths are far removed from most physical reality.

For many students, my job is to remind them to be very, very specific about language. What exactly does the question ask? What exactly is the definition of this term? How can we use the latter to approach the former? I wish there were more mathematician-writers, but the only decent one alive now that I can think of is Jordan Ellenberg. Maybe I can be next.

The rest of my time, right now, is taken up with some freelance work and job applications. A little reading, surfing, and python programming in spare moments.

Internet Topology:

2017-04-15 ↗

found image -

It's an understatement to say I have broad interests, but two of the most consistent threads (going on a decade here...) are in mathematics and visual art.

When this comes up in party conversation, people often ask how the two are related, or if I do art-inspired math or math-inspired art. My response to the first is that they both deal with expression and beauty/truth through symbols and abstraction. Do I blend of the two? Nope.

But why not? I've struggled to articulate the "why" of my answer to the second question, but much if it stems from finding most math-cum-art projects lacking in depth.

A statistic analysis of Bob Ross' paintings? Neat. But conceptually boring—Ross' paintings, while beautiful, don't have conceptual depth, and the statistics are vanilla.

I think George Hart's sculptures are rad (I got one installed at Brown), and they're good visualizations of mathematical shapes. But the beauty of the geometries are bogged down by presentation as folk sculpture, and the pieces are static, without emotional charge.

Nikki Graziano's Found Functions is a project that achieves something conceptually interesting as both math and art. The photographs are provocative, and the overlay excites further interpretation; the functions are simplistic but demonstrate a potential directness of mathematical modeling.

Why, though, do I still judge these two things separately, rather than being able to intertwine their evaluation?

I am finally rounding around to an answer to this question—why the two fields seem as difficult to blend effectively as oil and water. It comes back to the connection between the two—that they deal with expression through abstraction. The differing approaches to this abstraction, however, make blending difficult.

Canonical abstract art deals with nonrepresentation, or at least non-literal representation. (Abstraction is a spectrum, but the requirement here is an intentional aesthetic of inaccurate or non-real representation.) In an art context, an abstractly realized element can be interpreted to stand-in for something else—something more concrete for the audience, lower on the ladder of abstraction, or something more broadly defined, up on the ladder, to capture broad ideas. ("The woman represents the artist's mother" vs "the woman represents womanhood".)

The details of the implementation are generally taken as crucial: how does the line feel? what energy does this particular color field capture? what does this choice in plastic connote? Elements of visual art are necessarily instantiated in the world, and the details of that instantiation are interpreted by the viewer as key semiotic elements at play.

TL;DR In canonical abstract art, unique details in the instantiation of formal elements are taken as crucial to the meaning and interpretation. Visual artists mine the richness of realized objects to create meaning.

Math, however, moves in a nearly diametrically opposed direction. For a mathematician, there is a power in keeping forms in the abstract. The moment you draw a circle, it's an imperfect circle. But, if we refer to a circle only in its noninstantiated platonic form, it can remain perfect.

As mathematical objects remain uninstantiated, they stay exactly as we made them to be. Instantiation allows for chance of the physical world to enter the process, but a mathematical proof can remain tidy and general in scope without. While lots of mathematical interest comes out of unique cases, these are generally still cases of uniqueness at a high level of abstraction.

Instantiation of math objects brings it all crashing down to earth. When we take about the field, for example, Zmod7, we say the field rather than a field. If we actually collected a set of objects, and played out operations of the field with the objects, it would be a representation of the field. It might be a powerful learning tool, but it's an extra layer between our thinking and the object we'd actually like to manipulate. Instantiation of mathematical objects quickly feels trifling.

Graziano's piece pleasantly aestheticizes the typographic form of the function, despite the actual parts of the function being pretty arbitrary. This is an important point to note, I think—the notation we use is exactly the intersection of unique instantiation and platonic form. As a reader, we consider the platonic form by virtue of reading an instantiated symbol. (The same goes for visual art! Mathematicians just don't acknowledge this much. Most everything is automatically typeset by LaTeX and the author focuses on proof, as if it exists in a cultural vacuum. See also Ethnomathematics.)

For a project to meaningfully engage with both math and visual art, it needs to respect both the richness in instantiated detail, as well as the power of noninstantiated abstraction. This is a thin, thin tightrope.

For this reason, the "scores" of Fluxus and related movements are really engaging to me! They're like proofs, but for art-making. The power of the score is present to the reader, and the enaction of the piece serves as its context-dependent instantiation.

What other dimensions of abstraction are there? What's the characteristic difference between abstraction and simplification? How does nonrepresentationalism play into the metaphysics of mathematics? More to entertain, always. For now, though, this dimension of instantiation serves well as the distinction between these two fields I love.

Internet Topology:

2017-03-21 ↗

When I tell my friends from college that I moved to Los Angeles, most of them ask why in the world I choose to stay in L.A.

Coming off the East coast, an attraction to Los Angeles is not only unfashionable but also misunderstood. I've tried to touch on it before, and here I'll crowdsource some help.

It is difficult to capture the Los Angeles ethos in a photograph without falling to tropes or banality. Another strip mall? The mountains with palm trees in the foreground? The phenomenology of Los Angeles is woven from threads of of experience, rather than a patchwork of moments. It is the opposite of photogenic.

In my eyes, New York City is an amazing city to visit. Take the subway, eat a bagel, go to a museum, see a friend (or five). But it can suck to live in New York between the demanding social scene and fewer and fewer square feet per person. Conversely, Los Angeles is lovely to live in—regular sunshine, diverse topography, cheap food—but it sucks to visit. Half of your visit will be spent on the freeway, going between urban cores. It's tough for a visit to have any cohesion; the moments are banal.

L.A. Plays Itself, or, similarly, you play L.A. the way you want it. Geoff Manaugh captures some of this in his post Greater Los Angeles:

No matter what you do in L.A., your behavior is appropriate for the city. Los Angeles has no assumed correct mode of use. You can have fake breasts and drive a Ford Mustang – or you can grow a beard, weigh 300 pounds, and read Christian science fiction novels. Either way, you’re fine: that’s just how it works.

Los Angeles, the Improbably Sustainable City, makes no sense. I always felt like in the East Coast cities I know, the temperament of the city informs the stage on which you act. L.A. doesn't so much assert its own personality as much as it wears it with a wink while sitting in the seat next to you as you both watch the slow-motion apocalypse of Southern California.

K-Hole Description of L.A. K-Hole's Capture of the Paradox of L.A.

A lot of people think L.A. is just eyesore after eyesore. Full of mini-malls, palm trees, and billboards. So what?

So says Ice Cube in this short on L.A. and the Eames. He's right—so what?

"One man's eyesore is another man's paradise."

The individualism of the defining characters of Los Angeles—the creation of one's own paradise—cannot last. The ability for the city to act as a partner in finding one's own, glinting in the sunlight, might persist.

Internet Topology:

2017-02-27 ↗

How do you measure the density of a network?

In a simple, unweighted network, I immediately think of the traditional density metric: ratio of realized edges over possible edges. We can also entertain more nuanced values, like number of nodes in cliques over the total number of nodes. Even median path length is a decent measure of density. When we have edge weights that represent "length" or some such, possibilities expand.

These graph-theoretic networks can exist happily without worrying about spatial embedding. (Unless you want to get into that in a very precise way.) The whole point of graph isomorphism is that we want to think of graphs as isomorphic regardless of how they are drawn.

I now spend some time each week at Civic Projects, a planning consulting firm. We're working on a project that is all about overlapping networks in urban space. How does the network of bike paths commingle with the network of car-centric streets? How do networks of wayfinding systems intertwine with networks of lighting infrastructure?

These networks are much different from ones I was used to in college. They are inherently spatially embedded and human-centric. Isomorphism doesn't matter: these are networks with anecdotal, instantiated importance.

Urban infrastructure networks also range widely in type. If we want to talk about the "network" of parking infrastructure, raw count of parking spots per unit area is a decently descriptive metric. Node-based metrics aren't very good at representing the density of, say, a bicycle path network, because these metrics can collapse travel distances. Instead, we want something that captures the travel times and interconnectedness. In lighting, if we treat each source of light as a node in the network, some measure of bulk distance between elements helps grasp and what needs to be measured for best human use.

I'm coming around to three major types of urban networks. (Nothing novel here, just useful for classification.)

  1. Area-based networks. These take form as static resources. Parking, special zones (e.g. school zones), shade, and so forth.
  2. Path-based networks. These networks have intersections, but their experiential value emphasizes transit along edges rather than action at nodes. Travel paths populate this category.
  3. Node-based networks. These represent points of interaction. Lighting, gas stations, and wayfinding, all fall into this category.

Each of these necessitates a different strategy for density.

  1. Areal Densities. How many of something per unit area? What proportion of a given area is given to a specific form?
  2. Path Network Densities. This is, to my knowledge, a bit of uncharted territory. Metro's Active Transportation Strategic Plan advocates for an average distance between parallel elements in a network. (In the case of roadways, this is comparable to the width of city blocks.) But in which direction should we measure? Do we integrate along the length of the block? Do we measure the largest possible diameter of each "block"? We could, also, resort to average network path length per unit area (e.g. "how many miles of bike path are there per square mile?").
  3. Nodal Densities. These are closer to typical graph-theoretic density measurements, but should tend towards accounting for edge lengths as well. How far apart are the nodes, on average? How far apart are they from their nearest neighbors, given some threshold of "nearest" or some kind of partitioning?

Context-dependent answers are necessary to make effective, well-reasoned, and responsible metrics for these types. And sometimes, a particular street feature doesn't fit neatly into one category. Trash cans, for instance—we want to consider both an areal density, but also think carefully about the causes for litter—maybe line-of-sight to a receptacle—which falls more into a path density representation. We might need to explore metrics of both forms to make an intelligent decision regarding a proposed system.

Why is any of this useful?

Lots of planners want rules-of-thumb when pitching concepts and making cost estimations. Infrastructure network statistics can also help codify best practices for urban design.

Lots more research required in this area.

(colophon: I am now at SIAM CSE 2017 to run a workshop on social systems modeling, through the Broader Engagement program. Brain getting back into math mode.)

Internet Topology:

2017-02-09 ↗

The Rampant Wave playground.

I've started using the phrase "infrastructural moments" to label certain human experiences in the landscape. Infrastructure is all around us but is designed to go unnoticed; infrastructure shouldn't have "moments" as much as it enables timelines. But

There are specific instances of friction in systems which cause "moments" and reveal the infrastructure underneath. Is most often a negative friction—an annoyance, or something that doesn't happen as expected—but can range towards neutral or joyful too.

We expect our roads to be smooth, our streetlamps to function, our water to come out of the faucet. When we encounter an unanticipated road hazard, a dark street, and a gap in our water service, these are infrastructural moments that can trigger deep engagement with our infrastructure space.

I have collected some instantiations of "guerilla urbanism," which are very much examples of infrastructural moments. When individuals take the agency to seed-bomb an otherwise untended highway divider, it is a moment of recognition of the structures that are designed to maintain that space. Graffiti artists quickly learn the processes by which graffiti is erased or left.

Skateboarders are constantly mining urban infrastructure towards repurposing. Can this ADA-compliant ramp become my launching point? Can I get a ride in between police beat cycles? Skateboarding becomes a continuum of infrastructural moments, which informs an overhaul in outlook. Walk around a city with a skateboarder, and you won't see curbs as just curbs any more. Skateboarding is an entire approach to the built environment.

I walk East on Beverly Boulevard to work downtown. After a bridge over Glendale, Beverly becomes 1st St. On the Eastern side of the bridge, there's an on-ramp squeezed between some low-lying office buildings and the road. You can only access the ramp from an awkward turn at the intersection underneath the bridge. It's a very high-visibility place, but hard to access. In my 2+ months of doing this walk, there's been an old corvette with a flat tire on this ramp. Despite the parking spots along this ramp are labeled as 2-hour parking, the corvette has never had a ticket. Whether intentional or not, the owner seems to have found a gap in parking enforcement patterns. This is an infrastructural moment.

Another example: the ramp between the 110 South and 105 going West has a rate-limiting light. I approached the light one evening and saw it on at a distance, but no cars in the queue. As I neared, the light turned off completely; the flow was sparse enough that the light was no longer necessary. Did the power go out? Did I trigger something? What conditions turn the light off (it can't be manual, I would think)? I was present at the moment of an infrastructural adjustment—this is an infrastructural moment.

What other forms do infrastructural moments take? In what other contexts to they inform dispositions (ala the skateboarder)? Are there fundamental differences in moments of sublimity, frustration, and utility? How beneficial is it for these moments to be more common, more reflective?

Internet Topology:

My favs: Bit Player

2017-05-17 ↗ Life Update
2017-04-15 ↗ Math vs Visual Art
2017-03-21 ↗ Los Angeles Phenomenology
2017-02-27 ↗ Network Density
2017-02-09 ↗ Infrastructural Moments
2017-02-07 ↗ Learning Gardens Kickstarter
2017-01-15 ↗ Infrastructure as Intersection of Abstraction & Humanity
2016-12-21 ↗ Institutional Scales & Museums
2016-11-29 ↗ Surfing (and Learning Gardens)
2016-11-16 ↗ Dimensional Compression, Traffic, Chaos
2016-11-09 ↗ Trump, the Environment, Land Use, Heavy Headspace
2016-10-26 ↗ Transitions & Low-Margin Lifestyle
2016-10-14 ↗ New Geography, Los Feliz, Prioritization
2016-10-09 ↗ Leaving Twitter
2016-10-02 ↗ Manmade vs natural landscaping, conceptual art
2016-09-21 ↗ Internet topography and making publics
2016-09-20 ↗ HTML-native collage work
2016-09-18 ↗ Queues, surfing the internet, and surfing
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