Logic can only go so far
after that I must see-perceive-imagine.
This geometry can help.

I may reason logically thru theorem
and propositions galore,
but only what I perceive is real.

If after studying I am not changed
if after studying I still see the same

then all has gone for naught.

Geometry is to open up my mind
so I may see what has always been behind
the illusions that time
and space construct.

Space isn't made of point and line
the points and lines are in the mind.
The physicists see space as curved
with particles that are quite blurred.
And, when I draw, everything is fat
there are no points and that is that.
The artists and the dreamer knows
that space is where an image grows.
For me it's a sea in which I swim
a formless sea of hope and whim.

Thru my fear of Infinity and One
I structure space to confine
my imagination away from the idea
that all is One.

But, I can from this trap escape
I can see the geometry in which I wander
as but a structure I made to ponder.

I can dare to let go the structures
and my fears
and look beyond
to see what is always there to see.

But, to let go, I must first grab on.
Geometry is both the grabbing on
and the letting go.
It is a logical structure
and a perceived meaning

Q.E.D.'s and "Oh! I see!"'s.
It is formal abstractions
and beautiful contraptions.
It is talking precisely about that
which we know only fuzzily.
But, in the end, and, most of all,
it is seeing-perceiving

the meaning that

— David Henderson, 1978


In Euclidean, Spherical, and
Hyperbolic Spaces

Second Edition

David W. Henderson

Cornell University, Ithaca, New York


Daina Taimiða

University of Latvia, Rìga

Contributors to the First Edition
(Experiencing Geometry on Plane and Sphere)

Eduarda Moura

Justin Collins, Kelly Gaddis, Elizabeth B. Porter, Hal W. Schnee, Avery Solomon


Upper Saddle River, New Jersey 07458

Library of Congress Cataloging-in-Publication Data

Henderson, David W. (David Wilson), 1939

Experiencing geometry: in Euclidean, spherical, and hyperbolic spaces.--2nd ed./

David W. Henderson.

p. cm.

Includes bibliographical references and index.

ISBN 0-13-030953-2

1. Geometry. I. Title: Experiencing geometry in Euclidean, spherical, and hyperbolic

spaces. II. Title.

QA453 .H497 2001

516--dc21 00-037494

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All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher.

Portions of this material are based upon work supported by the National Science Foundation under Grant No. USE-9155873 and by Dwight David Eisenhower Title IIA grants administered by the New York State Department of Education. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation or the New York State Department of Education.

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To all the students who have studied geometry with me

(you have taught me much about geometry)

and to

Mount Ætna, the ùæúnìtis, and the angels