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For a decade and half, I have been spending hundreds of hours and
thousands of dollars each month composing The Marginalian ([13]formerly
Brain Pickings for its first fifteen years). It has remained free and
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Sunday newsletter

The Marginalian has a free Sunday digest of the week's most
mind-broadening and heart-lifting reflections spanning art, science,
poetry, philosophy, and other tendrils of our search for truth, beauty,
meaning, and creative vitality. Here's an [27]example. Like? Claim
yours:
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midweek newsletter

Also: Because The Marginalian is well into its second decade and
because I write primarily about ideas of timeless nourishment, each
Wednesday I dive into the archive and resurface from among the
thousands of essays one worth resavoring. Subscribe to this free
midweek pick-me-up for heart, mind, and spirit below — it is separate
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Also

[28]The Snail with the Right Heart: A True Story

[29]The Snail with the Right Heart: A True Story

[30]The Universe in Verse

[31]The Universe in Verse

[32]Figuring

[33]Figuring

[34]A Velocity of Being

[35]A Velocity of Being

[36]art

[37]sounds

[38]bites

[39]bookshelf

Favorite Reads

[40]Becoming the Marginalian: After 15 Years, Brain Pickings Reborn

[41]Becoming the Marginalian: After 15 Years, Brain Pickings Reborn

[42]Essential Life-Learnings from 14 Years of Brain Pickings

[43]Essential Life-Learnings from 14 Years of Brain Pickings

[44]Kinship: Ursula K. Le Guin’s Love Poem to Trees, the Interleaving
of Life and Death, and the Eternal Flame of Being

[45]Kinship: Ursula K. Le Guin’s Love Poem to Trees, the Interleaving of Life
and Death, and the Eternal Flame of Being

[46]Emily Dickinson’s Electric Love Letters to Susan Gilbert

[47]Emily Dickinson’s Electric Love Letters to Susan Gilbert

[48]Singularity: Marie Howe’s Ode to Stephen Hawking, Our Cosmic
Belonging, and the Meaning of Home, in a Stunning Animated Short Film

[49]Singularity: Marie Howe’s Ode to Stephen Hawking, Our Cosmic Belonging,
and the Meaning of Home, in a Stunning Animated Short Film

[50]How Kepler Invented Science Fiction and Defended His Mother in a
Witchcraft Trial While Revolutionizing Our Understanding of the
Universe

[51]How Kepler Invented Science Fiction and Defended His Mother in a
Witchcraft Trial While Revolutionizing Our Understanding of the Universe

[52]Hannah Arendt on Love and How to Live with the Fundamental Fear of
Loss

[53]Hannah Arendt on Love and How to Live with the Fundamental Fear of Loss

[54]The Cosmic Miracle of Trees: Astronaut Leland Melvin Reads Pablo
Neruda’s Love Letter to Earth’s Forests

[55]The Cosmic Miracle of Trees: Astronaut Leland Melvin Reads Pablo Neruda’s
Love Letter to Earth’s Forests

[56]Rebecca Solnit’s Lovely Letter to Children About How Books Solace,
Empower, and Transform Us

[57]Rebecca Solnit’s Lovely Letter to Children About How Books Solace,
Empower, and Transform Us

[58]Fixed vs. Growth: The Two Basic Mindsets That Shape Our Lives

[59]Fixed vs. Growth: The Two Basic Mindsets That Shape Our Lives

[60]In Praise of the Telescopic Perspective: A Reflection on Living
Through Turbulent Times

[61]In Praise of the Telescopic Perspective: A Reflection on Living Through
Turbulent Times

[62]A Stoic’s Key to Peace of Mind: Seneca on the Antidote to Anxiety

[63]A Stoic’s Key to Peace of Mind: Seneca on the Antidote to Anxiety

[64]The Courage to Be Yourself: E.E. Cummings on Art, Life, and Being
Unafraid to Feel

[65]The Courage to Be Yourself: E.E. Cummings on Art, Life, and Being
Unafraid to Feel

[66]The Writing of “Silent Spring”: Rachel Carson and the
Culture-Shifting Courage to Speak Inconvenient Truth to Power

[67]The Writing of “Silent Spring”: Rachel Carson and the Culture-Shifting
Courage to Speak Inconvenient Truth to Power

[68]A Rap on Race: Margaret Mead and James Baldwin’s Rare Conversation
on Forgiveness and the Difference Between Guilt and Responsibility

[69]A Rap on Race: Margaret Mead and James Baldwin’s Rare Conversation on
Forgiveness and the Difference Between Guilt and Responsibility

[70]The Science of Stress and How Our Emotions Affect Our
Susceptibility to Burnout and Disease

[71]The Science of Stress and How Our Emotions Affect Our Susceptibility to
Burnout and Disease

[72]Mary Oliver on What Attention Really Means and Her Moving Elegy for
Her Soul Mate

[73]Mary Oliver on What Attention Really Means and Her Moving Elegy for Her
Soul Mate

[74]Rebecca Solnit on Hope in Dark Times, Resisting the Defeatism of
Easy Despair, and What Victory Really Means for Movements of Social
Change

[75]Rebecca Solnit on Hope in Dark Times, Resisting the Defeatism of Easy
Despair, and What Victory Really Means for Movements of Social Change

[76]see more

Related Reads

[77]In Praise of Missing Out: Psychoanalyst Adam Phillips on the
Paradoxical Value of Our Unlived Lives

[78]In Praise of Missing Out: Psychoanalyst Adam Phillips on the Paradoxical
Value of Our Unlived Lives

[79]Love, Death, and Whitman: Poet Mark Doty on the Paradox of Desire
and the Courage to Love Against the Certitude of Loss

[80]Love, Death, and Whitman: Poet Mark Doty on the Paradox of Desire and the
Courage to Love Against the Certitude of Loss

[81]10 Learnings from 10 Years of Brain Pickings

[82]10 Learnings from 10 Years of Brain Pickings

Labors of Love

[83]Famous Writers' Sleep Habits vs. Literary Productivity, Visualized

[84][sleepproductivitywriters_500_1.jpg]

[85]7 Life-Learnings from 7 Years of Brain Pickings, Illustrated

[86][holstee_7things.jpg]

[87]Anaïs Nin on Love, Hand-Lettered by Debbie Millman

[88][anais_debbie1.jpg]

[89]Anaïs Nin on Real Love, Illustrated by Debbie Millman

[90][anaisnin_debbiemillman2_500.jpg]

[91]Susan Sontag on Love: Illustrated Diary Excerpts

[92][sontaglove_unlimited.jpg]

[93]Susan Sontag on Art: Illustrated Diary Excerpts

[94][sontagart.jpg]

[95]Albert Camus on Happiness and Love, Illustrated by Wendy MacNaughton

[96][wendycamus.jpg]

[97]The Holstee Manifesto

[98][holsteemanifesto.jpg]

[99]The Silent Music of the Mind: Remembering Oliver Sacks

[100][oliversacks_debbiemillman.jpg]

The Geometry of Grief: A Mathematician on How Fractals Can Help Us Fathom
Loss and Reorient to the Ongoingness of Life

“The distance between here and there is the answer to the wrong question.”

By Maria Popova

[101]The Geometry of Grief: A Mathematician on How Fractals Can Help Us
Fathom Loss and Reorient to the Ongoingness of Life

“What exists, exists so that it can be lost and become precious,” Lisel
Mueller wrote in her [102]stunning poem about what gives meaning to our
mortal lives as she neared, but never quite reached, the triumph of
having lived a century — a bittersweet triumph, for to live at all,
however long or short, is an unbidden bargain to lose everything you
hold precious: every love and every life, including your own. Loss is
the price of life — a price we never chose to pay any more than we
chose to be born, and yet a price not merely worth paying but beyond
questions of worth and why.

One corollary is that, both in the evolutionary sense and in the
existential, [103]every loss reveals what we are made of. But every
loss also reveals what it is made of, which is more loss: Each loss
takes a piece of us — a piece soft and alive — and leaves in its place
something cold and heavy; each subsequent loss becomes the magnet that
draws out those old leaden pieces, pulls them out from the reliquary of
scar tissue where we have been keeping them in order to live, makes
them rip through our being afresh. And yet the shrapnel pieces that
surface are smaller and softer-edged than when they first entered
through the open wound of raw bereavement, smoothed and contracted by
the ongoingness of life.

In this sense, grief if fractal, each new instance containing within
itself a set of self-similar sub-griefs — miniatures of the same
emotional structure, rendered smaller in salience by time and tenacity,
those twin inevitabilities of aliveness.
The Mandelbrot set. (Illustration by Wolfgang Beyer.)

How the fractal nature of grief is both the key to understanding it and
the doorway to moving through it is what mathematician Michael Frame
explores in his unusual book [104]Geometry of Grief: Reflections on
Mathematics, Loss, and Life ([105]public library). After twenty years
of working with [106]the visionary father of fractals and another
twenty years of teaching fractal geometry at Yale, Frame draws on a
lifetime of loss and a lifetime of delicate attention to the details of
aliveness we call beauty to interleave memoir and mathematics in an
uncommon tapestry of thought, twining Borges and quantum mechanics,
evolutionary biology and Islamic art, music and multiverse theory.

Because every sound theorem rests upon precise formulation, Frame
offers a basic definition:

Grief is a response to an irreversible loss… To generate grief
rather than sadness, the thing lost must carry great emotional
weight, and it must pull back the veil that covers a transcendent
aspect of the world. Breathe out to push the fog away from a
brilliant pinpoint of light.

[107]Total eclipse of the sun, observed July 29, 1878, at Creston,
Wyoming Territory Total eclipse of 1878, one of Étienne Léopold
Trouvelot’s [108]groundbreaking astronomical drawings. (Available
[109]as a print and [110]as a face mask.)

This trifecta of irreversibility, emotional heft, and transcendence
anchors Frame’s model of grief and his map for navigating the landscape
of loss not as a journey of recovery but as one of readjustment — of
reconstituting our model of the world within, which governs our entire
experience of the world without. Because the two basic building blocks
of our world-model — inner and outer — are [111]attention and
[112]narrative, readjustment to life after loss requires deliberate
wielding of both. Frame writes:

All moments of our lives are immensely rich, with many — perhaps
infinitely many — variables we could notice.

We can view our lives as trajectories, parameterized by time,
through story space.

We can never simultaneously view all of the possible variables;
rather, we focus on a few variables at a time, restricting our
attention to a low-dimensional subspace of story space.

Our trajectories through these subspaces are the stories we tell
ourselves about our lives; they are how we make sense of our lives,
but always they miss some elements of our experiences.

Irreversible loss appears as a discontinuity, a jump, in our path
through story space.

By focusing on certain subspaces, by projecting our trajectories
into these spaces, we can reduce the apparent magnitude of the
jumps, and consequently find a way to confront the emotional loss
and perhaps reduce its impact.

The most gladdening thing about grief parallels the most gladdening
thing about science: However meticulous our projections and our models
of reality may be, however triumphant in their conquest of knowledge,
they are not only perennially incomplete but could be — and, throughout
the history of our species, have often been — fundamentally wrong.
Science, like life itself, rests upon [113]the abstract art of
otherwise — things could be other than what they appear to be, other
than what we assume them to be: stranger, more slippery, more possible.
Frame writes:

Geometry is a way to organize our models of the world, its shapes
and dynamics. But isn’t this all contingent, balanced on a knife’s
edge? Could our models have turned out very differently? If the
fractal geometry of Mandelbrot had been discovered before the
geometry of Euclid, would manufacture be the same? If you think the
question is far-fetched, consider the iterated branching of our
pulmonary, circulatory, and nervous systems, or the recursive
folding of our DNA, or the large surface area and small volume of
our lungs and our digestive tract. Evolution has discovered and uses
fractal geometry. If people had looked more closely at the geometry
of nature, rather than emulating the “celestial perfection” imposed
by the church’s interpretation of the works of Euclid and Aristotle,
our constructions could be very different now.

Solids from Kepler’s Harmony of the World, exploring the relationship
between harmony and geometry. (Available [114]as a print and [115]as a
face mask.)

To be fair, the rare few did look and did see different constructions
of reality — the [116]Hungarian teenager who, two hundred years ago,
subverted Euclid and equipped Einstein with the building blocks of
relativity; the sickly German mathematician who, four hundred years
ago, subverted the celestial interpretations of the church to give us
the revolutionary laws of planetary motion [117]while defending his
mother in a witchcraft trial.

Frame writes:

Unless there were only one geometry, only one story — only one world
— we should not expect the same categories to grid our views of the
universe… Could the world be different than we think? Is it
different? Must it be only one thing, or can it be many? If we view
the world in one way, does this forever bar us from all others?

Art from [118]An Original Theory or New Hypothesis of the Universe by
Thomas Wright, 1750. (Available as [119]a print, as a [120]face mask,
and as [121]stationery cards.)

Pointing to a resounding “no” in the many-worlds model of quantum
mechanics — a model in which “every observation of every particle
splits the universe into branches, one in which each measurement
outcome occurs, and communication between these branches is impossible”
— he adds:

Once they are recognized, these patterns cannot go unnoticed. They
change forever how the image of the world unfolds in our minds,
change forever the categories of the models we build.

This recognition-as-model-revision, Frame intimates, is also the way to
view and live through grief — an exercise in continual dilation of
perspective, so that life can be seen from more and more angles besides
the acuteness of loss, noticing more and more of what is there, what
remains and what grows in the wake of the lost; an exercise in
remembering, again and again, that healing is subtle and unpredictable,
unfolding in tiny, quiet, immeasurable increments that eventually add
up to profound changes of measurable difference.

Returning to the consolation of fractals — the mathematical language
composing chaos theory — Frame writes:

Small changes may not cause large differences, but small changes,
invisible because of our inability to measure exactly, can mask our
ability to predict whether, when, and where large differences can
occur. Chaos is about the breakdown of our ability to forecast for
more than a short time.

[122][wilsonbentley_snowflakes22.jpg?w=1029&ssl=1] One of Wilson
Bentley’s [123]pioneering 19th-century photographs of snowflakes, one
of nature’s fractal masterpieces.

What most readily unblinds us to that vital smallness comprising the
grandeur of change and aliveness is a willful attentiveness to beauty —
so often [124]the antipode to the brutality of life, so often [125]the
portal to aliveness in the face of death, always the supreme testament
to pioneering psychologist and philosopher William James’s insight that
[126]our experience is what we choose to attend to.

Attentiveness to beauty is the instrument of transcendence — that
essential facet of Frame’s geometry of grief and readjustment. In
consonance with Willa Cather’s lovely insistence that [127]“unless you
can see the beauty all around you everywhere, and enjoy it, you can
never comprehend art” — or life — he writes:

Beauty is a bridge between grief and geometry.

[…]

Beauty and grief are next-door neighbors, or maybe grief is beauty
in a dark mirror… To see beauty is to glimpse something deeper; to
grieve is to glimpse a loss whose consequences we will not unpack
for years, and maybe never. The beauty of geometry likewise involves
great emotional weight, irreversibly alters our perceptions, and is
transcendent. For we don’t see all of geometry, only a hint, a
shadow of something much deeper.

The Dreaming Horses by [128]Franz Marc, 1913. (Available [129]as a
print and as [130]stationery cards.)

In one of the book’s tenderest moments, illustrating this sidewise
gleam at the depths, Frame shares a short lyrical essay he composed
after his mother’s death, in response to a creative prompt from a
student compiling meditations on gravity:

Gravity holds my feet on the ground. Gravity keeps the earth
traveling around the sun, the sun dancing around the galaxy, the
galaxy threading through the Local Group, and on and on.

Gravity pulls rain out of the sky. And snowflakes. And leaves in
autumn. And tears from my eyes when I knew you really are gone.
Where did you go?

[…]

The distance between here and there is the answer to the wrong
question.

[…]

I thought gravity pulled my mind into the past, stuck in memories.
But now I know I can’t trust memories. Some are invented, all are
edited. The whole web of who I am — what I’ve seen and done, what
skills I’ve found — is nothing but fog.

Gravity pulls me to the future, bits of me falling off along the
way. Each of us disappears into the mist of the possible. In our
minds, time is gravity’s other side.

Complement Frame’s [131]Geometry of Grief with Emily Dickinson (who
believed that “best witchcraft is geometry”) on [132]the dual spell of
love and loss, Hannah Arendt on [133]the antidote to the
irreversibility of life, Derek Jarman on [134]gardening as a means of
growing though grief, and Nick Cave on [135]loss as a portal to
aliveness, then revisit the story of how Benoit Mandelbrot’s discovery
of fractals illuminated [136]the hidden order behind chaos.

donating = loving

For a decade and half, I have been spending hundreds of hours and
thousands of dollars each month composing The Marginalian
([137]formerly Brain Pickings for its first fifteen years). It has
remained free and ad-free and alive thanks to patronage from readers. I
have no staff, no interns, no assistant — a thoroughly one-woman labor
of love that is also my life and my livelihood. If this labor makes
your own life more livable in any way, please consider lending a
helping hand. Your support makes all the difference.

Monthly donation

♥ $3 / month

♥ $5 / month

♥ $7 / month

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Sunday newsletter

The Marginalian has a free Sunday digest of the week's most
mind-broadening and heart-lifting reflections spanning art, science,
poetry, philosophy, and other tendrils of our search for truth, beauty,
meaning, and creative vitality. Here's an [141]example. Like? Claim
yours:
____________________
____________________
____________________
Subscribe

midweek newsletter

Also: Because The Marginalian is well into its second decade and
because I write primarily about ideas of timeless nourishment, each
Wednesday I dive into the archive and resurface from among the
thousands of essays one worth resavoring. Subscribe to this free
midweek pick-me-up for heart, mind, and spirit below — it is separate
from the standard Sunday digest of new pieces:
____________________
____________________
____________________
Subscribe

—
Published November 10, 2021
—
https://www.themarginalian.org/2021/11/10/geometry-of-grief-michael-fra
me/
—
BP

www.themarginalian.org

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58. https://www.themarginalian.org/2014/01/29/carol-dweck-mindset/
59. https://www.themarginalian.org/2014/01/29/carol-dweck-mindset/
60. https://www.themarginalian.org/2017/12/21/reflection/
61. https://www.themarginalian.org/2017/12/21/reflection/
62. https://www.themarginalian.org/2017/08/27/seneca-anxiety/
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69. https://www.themarginalian.org/2015/03/19/a-rap-on-race-margaret-mead-and-james-baldwin/
70. https://www.themarginalian.org/2015/07/20/esther-sternberg-balance-within-stress-emotion/
71. https://www.themarginalian.org/2015/07/20/esther-sternberg-balance-within-stress-emotion/
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91. https://www.themarginalian.org/2012/08/03/susan-sontag-on-love/
92. https://www.themarginalian.org/2012/08/03/susan-sontag-on-love/
93. https://www.themarginalian.org/2012/11/30/susan-sontag-on-art/
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101. https://www.amazon.com/Geometry-Grief-Reflections-Mathematics-Loss/dp/022680092X/?tag=braipick-20
102. https://www.themarginalian.org/2020/02/24/immortality-in-passing-lisel-mueller/
103. https://www.themarginalian.org/2021/10/26/why-leaves-change-color/
104. https://www.amazon.com/Geometry-Grief-Reflections-Mathematics-Loss/dp/022680092X/?tag=braipick-20
105. https://www.worldcat.org/title/geometry-of-grief-reflections-on-mathematics-loss-and-life/oclc/1237632211&referer=brief_results
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107. https://www.themarginalian.org/2016/07/07/trouvelots-astronomical-drawings/
108. https://www.themarginalian.org/2016/07/07/trouvelots-astronomical-drawings/
109. https://society6.com/product/total-solar-eclipse-by-tienne-lopold-trouvelot-1878-et5_print#s6-4686076p4a1v1?curator=brainpicker
110. https://society6.com/brainpicker/collection/vintage-science-face-masks?curator=brainpicker
111. https://www.themarginalian.org/2016/03/25/william-james-attention/
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114. https://society6.com/product/art-from-the-harmony-of-the-world-by-johannes-kepler-1619_print?sku=s6-19311111p4a1v46?curator=brainpicker
115. https://society6.com/brainpicker/collection/vintage-science-face-masks?curator=brainpicker
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117. https://www.themarginalian.org/2019/12/26/katharina-kepler-witchcraft-dream/
118. https://www.themarginalian.org/2021/02/16/thomas-wright-original-theory/
119. https://society6.com/product/art-from-thomas-wrights-an-original-theory-or-new-hypothesis-of-the-universe-1750_print?curator=brainpicker
120. https://society6.com/brainpicker/collection/vintage-science-face-masks?curator=brainpicker
121. https://society6.com/brainpicker/collection/vintage-science-cards?curator=brainpicker
122. https://www.themarginalian.org/2020/01/19/wilson-bentley-snowflakes/
123. https://www.themarginalian.org/2020/01/19/wilson-bentley-snowflakes/
124. https://www.themarginalian.org/2021/09/08/franz-marc/
125. https://www.themarginalian.org/2020/04/16/mary-shelley-the-last-man/
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128. https://www.themarginalian.org/2021/09/08/franz-marc/
129. https://society6.com/product/the-dreaming-horses-by-franz-marc-1913_print?curator=brainpicker
130. https://society6.com/brainpicker/cards?curator=brainpicker
131. https://www.amazon.com/Geometry-Grief-Reflections-Mathematics-Loss/dp/022680092X/?tag=braipick-20
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133. https://www.themarginalian.org/2021/07/14/hannah-arendt-forgiveness/
134. https://www.themarginalian.org/2021/04/04/derek-jarman-modern-nature-gardening/
135. https://www.themarginalian.org/2020/07/27/nick-cave-loss-grief/
136. https://www.themarginalian.org/2021/02/22/mandelbrot-fractals-chaos/
137. https://www.themarginalian.org/2021/10/22/brain-pickings-becoming-the-marginalian/
138. https://www.themarginalian.org/donate/
139. https://www.themarginalian.org/donate/
140. https://www.themarginalian.org/cancel-recurring-donation/
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157. https://society6.com/product/art-from-thomas-wrights-an-original-theory-or-new-hypothesis-of-the-universe-1750_print?curator=brainpicker
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