Monthly Archives: February 2025

Waiting for the Music

I am working on a post focused on eight (by one method of counting) intertwining mathematical threads. At the present moment, a significant part of my attention is centered on exposing the simplicity and beauty and connection that become obvious when you are immersed in the experience of the mathematical poetry these threads illuminate.

The eight pieces are (order not significant) the area formula, the Gauss map, curvature measures, Gaussian curvature, degree theory, Gauss-Bonnet theorem, mean curvature, and the tube formulas.

I suppose the key feature of this quest for the somewhat elusive music I hear when immersed, is the persistent conviction that there is some way to share this experience using (relatively) simple language and tools. This belief is founded in the experience (over and over) of the deeper, core essence of something that might at first look complicated, being simple, easily graspable and possible to explain to non-experts.

Coming back to the eight threads, the three core components are (a) the derivative as a linear approximation and the difference between the derivative and its determinant, (b) how sets and measures transform under maps with derivatives, and (c) three technical tools/ideas – Sard’s Theorem, regular level sets, degree theory – that enable us to get where we want to go, and illuminate how we get there. Central to this weaving together of the threads are the derivatives of two different maps – the normal flow generated by a surface with positive reach and the Gauss map of a co-dimension 1 submanifold of \(\Bbb{R}^n\) .

And as I am writing this, it seems to me that a significant part of the beauty is the way in which non-trivial conclusions emerge from the arrangement and collaboration of (a)-(c). Another piece is surely the sense of vital flow pervading the geometric, dynamic experience of seeing and feeling the mathematical realities formally exposed in the various theorems and lemmas. As I have written elsewhere, there is a language that cannot be spoken that is being evoked, triggered in the minds eye, in the heart that can hear those things.

While this might seem much too fuzzy and imprecise to some who read it, I suspect that there are quite a number of mathematical travelers that will resonate with those sentiments.

I believe that the best I can do in opening to others what I see and hear is to write as simply as possible, to not obscure the connections that contain the flow I refer to above, and to explicitly invite the reader to consider that what they are to get from what is written is much more than is written and is only visible when what is written has been internalized, converted to that language that cannot be spoken.

And so, what is written, at its best, becomes a trigger for an experience much greater that what is on the page, for a living, bare-handed exploration engaging the whole person. This is the goal to work towards.

Comment on “In the Company of Bees”

Of the previous five posts, two — “In the company of Bees (Again)” and “In the Company of Bees (and Again)” — were edits of the fifth post, “In the Company of Bees”.

I decided that instead of simply editing the original post, I would edit in full view. So I left versions 1 and 2 so that version 3 could be compared with them.

In general, I think too much polished product and too little process is visible in creative output, whether in mathematics papers, in poetry, in music, in film, etc. etc. While it is starting to be possible to get a look behind the scenes of large budget items like movies, and it is true that Youtuber DIYers sometimes post the full history of a project, it is still the case that creative output very often seems almost magical and not subject to struggle. This is true even if, in some rational, abstract way, we know a great deal of work and struggle was involved. Because we don’t feel the struggle, we can’t grasp the reality of the struggle.

So I thought I would experiment with having all three versions up there. Rubin’s book “The Creative Act: A Way of Being” played some sort of influence here. In particular, just before doing these two edits I read and felt “The Abundant Mindset”, “The Experimenter and the Finisher”, and “Temporary Rules”. I recommend Rubin’s book very highly.

In the Company of Bees (and Again)

A bee at work

Warmed by the sun, surrounded by bees focused on tiny flowers, the artificial, the disconnected, the un-grounded, fades away.

Occasionally, bees sit on my finger as they work. The smell of herbs, the sound of bees, the warm, quiet light, fill my attention. I enter an illuminated world outside the modern flow of time.

The bees and I commune, holding quietness together, inside and outside of time.

 

In the Company of Bees (Again)

A bee at work

Sitting quietly, surrounded by bees focused on tiny flowers, the artificial, the disconnected, the un-grounded fades away.

The sun’s warmth encompasses me.

Occasionally, bees sit on my finger as they work. The smell of the herbs, the sound of the bees, the undisturbed quietness, the light and warmth, combine to fill my attention, to become an illuminated world outside of a modern flow of time.

The bees and I commune, holding quietness together, inside and outside of time.

 

The Inner Quietness of Snow 

a silence falling, filled with light
crisp
sharply fresh
sunlight, dancing its way through the clouds

a flowing stillness
the white light of tiny crystals singing together

quiet poetry
moving light
the poetry of sunlight falling gently down

the inner quietness of snow 
illuminates everything

The Stillness We Inhabit 

stillness and words,
music from stillness

time stops, listening
 
stillness overflows

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the Stillness we inhabit
sings to us
words creating life
speaking the language that cannot be spoken