Table Of ContentGeometry of Grief
Geometry of Grief
REFLECTIONS ON MATHE MATICS,
LOSS, AND LIFE
Michael Frame
The University of Chicago Press
Chicago and London
The University of Chicago Press, Chicago 60637
The University of Chicago Press, Ltd., London
© 2021 by Michael Frame
All rights reserved. No part of this book may be
used or reproduced in any manner whatsoever
without written permission, except in the case of
brief quotations in critical articles and reviews. For
more information, contact the University of Chicago
Press, 1427 E. 60th St., Chicago, IL 60637.
Published 2021
Printed in the United States of America
29 28 27 26 25 24 23 22 21 20 1 2 3 4 5
ISBN- 13: 978- 0- 226- 80092- 9 (cloth)
ISBN- 13: 978- 0- 226- 80108- 7 (e- book)
DOI: https://doi.org/10.7208/chicago
/9780226801087.001.0001
Library of Congress Cataloging-in-Publication Data
Names: Frame, Michael, author.
Title: Geometry of grief : reflections on
mathematics, loss, and life / Michael Frame.
Description: Chicago : University of Chicago Press,
2021. | Includes bibliographical references and
index.
Identifiers: LCCN 2021007566 |
ISBN 9780226800929 (cloth) |
ISBN 9780226801087 (ebook)
Subjects: LCSH: Fractals. | Grief. | Geometry. |
Mathematics—Social aspects.
Classification: LCC QA614.86 .F796 2021 |
DDC 514/.742—dc23
LC record available at https://lccn.loc
.gov/2021007566
♾ This paper meets the requirements of
ANSI/NISO Z39.48- 1992 (Permanence of Paper).
Contents
Prologue 1
1: Geometry 15
2: Grief 46
3: Beauty 70
4: Story 92
5: Fractal 110
6: Beyond 126
Appendix: More Math 131
Acknowledgments 145
Notes 151
Index 163
Prologue
Dad, that’s really scary.
“Look for the very brightest one in the sky.”
“Beside the tree, about halfway up? Is that it, Ruthie?”
“That’s it. That’s Venus. It’s a planet, a whole world, almost
as big as the earth. And it’s cloudy all the time. No one has ever
seen land on Venus.”
“If it’s cloudy all the time, Venus must be cold.”
“Not necessarily. Venus is closer to the sun than Earth is.
Maybe the clouds hold in the heat and it’s very hot there.”
“Oh, I see. The sky is clear tonight, so we’ll get cooler than we
would if it was cloudy.”
“That’s right, Mikey. Do you want to go inside now?”
“Are there other planets in the sky?”
“Not tonight.”
“Can we stay outside and watch lightning bugs?”
“Sure.”
This was an evening late in the summer of 1958. The sky,
purple deepening to indigo, showed a few pinpoint stars and
a much brighter dot, Venus. We’d had dinner with my grand-
mother and my Aunt Ruthie, my dad’s sister, at their house in
South Charleston, West Virginia. I was seven, my sister Linda
was four, our brother Steve was two. Only Ruthie and I were in
the backyard. The others were on the front porch, “visiting,”
2 PROLOGUE
Mom called it. We lived in St. Albans, West Virginia, only about
eight miles away, and saw my grandmother and Ruthie often.
Just why the adults visited wasn’t clear to me. What could they
talk about? They just gossiped about their neighbors and other
family members.
Ruthie and I were different. That afternoon we’d sat in the
kitchen garden, and the purposeful march of ants and the
random jumps of grasshoppers entranced us. I constructed
elaborate natural histories to explain their behaviors; Ruthie
proposed much simpler alternatives. She never used the term
“Occam’s razor,” but she had begun to teach me the beauty of
simple explanations. And also the likelihood of economy: a Rube
Goldberg machine—a complicated contraption that takes up a
whole room and performs a simple task like cracking an egg—
has many points of potential failure. My complex pathways were
good mental exercises, maybe, but did I really think nature would
be that silly? Years later, I understood that Ruthie had started
me on the path to becoming a scientist. She thought curiosity
is the most important trait of the mind; that the curiosity of a
child, the twists and turns of young logic when the child unpacks
aspect and dynamic of the wide world, is the most beautiful thing
an adult can see. Mom and Dad, grandparents, other aunts and
uncles, encouraged curiosity, but Ruthie cultivated it, mixed in
some skepticism, and always found a book for me to read about
the topic of current interest. Ruthie set me on the way that, sixty
years later, has led me to write this story.
In elementary school career discussions, against my class-
mates’ police officer, firefighter, and park ranger (astronaut
wasn’t a career then—yes, I’m ancient), I offered physicist or
mathematician or astronomer. But really, at that age every
kid is a naturalist. A summer morning in neighboring woods
revealed wonders without end. The optimism of childhood
knew no bounds. My parents’ finances, though limited, afforded
PROLOGUE 3
opportunity for creative explorations. To measure the output of
a thermocouple (a copper wire and a steel wire twisted together
that convert heat into a weak electrical current), the father
of another student bought an expensive multimeter. I made a
galvanometer: two magnetized needles stuck through a small
cardboard rectangle suspended by thread in a coil of wire. Who
had more fun detecting the tiny current?
Ruthie didn’t help me design the experiments—Dad did that,
and let me set up a small lab in a corner of his workshop—but
Ruthie helped me realize that I could do experiments and answer
some of my own questions.
Late in my eleventh year, Ruthie got sick. Hodgkin’s lym-
phoma, survivable now but not so much in the early 1960s. She
was treated, with the chemotherapy drug Mustargen, I believe,
but lived only a few more months in some misery and died early
in my twelfth year. I visited Ruthie when she was sick, but I
couldn’t do much. I stood beside her bed, rested my little hand
on her forearm and tried to talk with her. But I couldn’t think
of anything to say. At home after these visits, Mom hugged me,
stroked my hair. I knew I should have talked more with Ruthie.
She had done so much for me, and she needed me now. She
needed me to talk with her because I was her favorite. Later I
understood that Mom was working through her own grief. She
knew the situation far better than I did, knew this disease would
win and Ruthie would lose. Dad began to talk with me about his
sister’s illness. He was straightforward: Ruthie was going to die.
I appreciated his honesty. No nonsense about Ruthie going away,
or—worse—going to live with the angels.1 Her life would end, and
soon. “This isn’t fair. There’s so much more for Ruthie and me to
do. She promised we’d get a telescope to look at the planets. I’ve
saved my allowance for six months already. This just isn’t fair.”
“Son, life isn’t fair. Ruthie isn’t sick because she did anything
bad. She just got sick. Sometimes good things happen, some-
4 PROLOGUE
times bad things happen. All we can do is try to make a few more
good things happen and a few less bad things happen. But a lot
of things that happen to us, we can’t do anything about.”
“Dad, that’s really scary.”
“Yes, son, it really is.”
That night I thought of a plan. I’d work very, very hard. Study
all the time, no more hide- and- seek or silly stories told to little
kids. I’d finish high school years early, go to college, then graduate
school and medical school, become a medical researcher, find a
cure for Hodgkin’s lymphoma, administer it to Ruthie, and save
her. In one version of the fantasy, I flew in a helicopter from my
university laboratory to Ruthie’s hospital. I was so pleased with
my plan. I told Mom and said I’d tell Ruthie not to worry, that I’d
save her. I expected Mom to be happy, but she looked very sad,
told me I couldn’t tell Ruthie.
“Why not? Don’t you want her to know she’ll be alright?”
“Mikey, I don’t want you to get her hopes up.” A lie, but a
gentle, sweet lie. “No matter how hard you work, you might not
be able to save Ruthie.”
Logically I knew Mom was right. I’d gone to the library in
Charleston, found an oncology book (I’d asked Mom the scien-
tific name for the study of cancer), and found the Hodgkin’s sur-
vival statistics. They weren’t encouraging. But I wasn’t able to
imagine a world without Ruthie. We had years of exploration
still to do. And besides, how could Ruthie leave her sweet mother,
Luverna Frame, the kindest, gentlest adult in my world? There
had to be a way out of this, and I would find it.
But Ruthie died. Dad was at the hospital with her, holding
her hand, when she died. When he came home, his expression
told me all I needed to know. He told Mom, Linda, and Steve.
They cried; I didn’t. Eventually Mom said that Ruthie had been
terribly sick, would never be well again, so it was better that she
wasn’t suffering anymore. “Ruthie was suffering?” Linda wailed.
