Woodcarving Magic_cutdown

WOODCARVING

MAGIC

HOW TO TRANSFORM A SINGLE BLOCK OF WOOD INTO IMPOSSIBLE SHAPES

WITH 29 PROJECTS FOR CARVING INTERLOCKING RINGS, CAGES & OTHER AMAZING GEOMETRIC DESIGNS

BJARNE JESPERSEN

“This beautiful and extraordinary book not only teaches, but inspires and awes.” – Ernö Rubik, creator of the Rubik’s Cube

Woodcarving

Magic

HoW to transform a single Block of Wood into impossiBle sHapes

With 29 Projects for carving interlocking rings, cages & other aMazing geoMetric Designs

Bjarne jesPersen

For Poul, who got me started, and Anne and Lene, who kept me going.

© 2012 by Bjarne Jespersen and Fox Chapel Publishing Company, Inc., East Petersburg, PA. Woodcarving Magic is an original work, first published in 2012 by Fox Chapel Publishing Company, Inc. The patterns contained herein are copyrighted by the author. Readers may make copies of these patterns for personal use. The patterns themselves, however, are not to be duplicated for resale or distribution under any circumstances. Any such copying is a violation of copyright law.

ISBN 978-1-56523-523-6 Library of Congress Cataloging-in-Publication Data Jespersen, Bjarne.

Woodcarving magic / Bjarne Jespersen. -- 1st. p. cm.

Includes index.

ISBN 978-1-56523-523-6 (alk. paper) 1. Wood-carving--Patterns. I. Title.

TT199.7.J48 2012 736’.4--dc23 2011021640 To learn more about the other great books from Fox Chapel Publishing, or to find a retailer near you, call toll-free 800-457-9112 or visit us at www.FoxChapelPublishing.com.

Note to Authors: We are always looking for talented authors to write new books in our area of woodworking, design, and related crafts. Please send a brief letter describing your idea to Acquisition Editor, 1970 Broad Street, East Petersburg, PA 17520.

Printed in China First printing Because working with wood and other materials inherently includes the risk of injury and damage, this book cannot guarantee that creating the projects in this book is safe for everyone. For this reason, this book is sold without warranties or guarantees of any kind, expressed or implied, and the publisher and the author disclaim any liability for any injuries, losses, or damages caused in any way by the content of this book or the reader’s use of the tools needed to complete the projects presented here. The publisher and the author urge all readers to thoroughly review each project and to understand the use of all tools before beginning any project.

acknowledgements This book would not have existed if I had

this book enjoyable reading. Lindsay Hess

not seen a small note from Fox Chapel

and Jason Deller designed this beautiful book

Publishing Company in David Springett’s

as if they could read my dreams, while Kerri

book Woodturning Wizardry, advertising for

Landis kept them from being disturbed by

authors who can write about woodworking.

tying up all loose ends. Several people and

I am grateful that this kind of publisher

institutions also helped by providing images

exists. My gratitude increased immensely

for Chapter 1; my thanks to them all.

when Fox Chapel enthusiastically greeted

My trip to the U.S. would not have

my proposal for this book.

These people deserve special thanks:

happened without a very generous grant from The David Collection in Copenhagen.

Acquisition Editor Peg Couch, for making

I am grateful to them. The grant also allowed

me feel wanted from the very beginning,

me to visit two fellow artists and soul mates

and everyone else at Fox Chapel who

I knew only from the Internet: Bathsheba

welcomed my wife and me so warmly when

Grossman, in Santa Cruz, California, and

we visited to have my work photographed.

George Hart, in Stony Brook, New York. My

The photographer, Scott Kriner, made my

meeting with George opened a new chapter

carvings look almost as if you could feel the

in my life because he got me involved with

curves and wood grain; he alone made my

Bridges, an organization that arranges annual

journey worthwhile. David Heim, my new

conferences on the connections between art

friend and text editor, did not just make

and mathematics. I am very glad that George

my text sound native. He also carved away

agreed to write the foreword.

a good deal of waste, sanded off all the

Finally, I want to thank my wife Lene for

bumps, and polished it to a silky shine that I

her patience and support during the months

had not imagined possible. In short, he made

I have been working on this book.

contents chaPter 4

about this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 about the author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

Mock-ups and Prototypes

. . . . . . . . . . . . .

40

chaPter 1

Pipe Cleaners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Metal Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 Copper Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 3-D Printing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

traditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

chaPter 5

chaPter 2

Preparing Blanks to carve . . . . . . . . . . . . . 46

Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Desirable Grain and Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 Finding Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 Drying Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 chaPter 3

tools and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Initial Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 Carving Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Carving Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Separating Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35 Final Shaping and Rounding Off . . . . . . . . . . . . . . . . . . . . . 37 Sanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 Surface Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

The Cube (or Bar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 The Cuboctahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48 Dodecahedron from a Cube . . . . . . . . . . . . . . . . . . . . . . . . . 50 Rhombic Dodecahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Dodecahedron Cut from a Board . . . . . . . . . . . . . . . . . . .52 chaPter 6

getting started

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

The Hopf Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 Lovers’ Hearts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Borromean Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

chaPter 7

flat rings

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Four Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 Square Dance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Six Pentagons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 Star Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Clover Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Halo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 chaPter 8

cages

76

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Octoball and Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Double Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Memento Mori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Flexus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 Asalink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84 Hosohedral Hexalink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85 Hosohedral Decalink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86 Five Inverted Tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87 chaPter 9

twisted rings

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Triple Whitehead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Tetracoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94 Helicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95 Hexacoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 Quintuple Borromeo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

c h a P t e r 10

knotted rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Small Tetraknot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 Great Tetraknot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Inverted Great Tetraknot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Small Concentric Tetraknot . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Great Concentric Tetraknot . . . . . . . . . . . . . . . . . . . . . . . . . 104 chaPter 11

creative geometry . . . . . . . . . . . . . . . . . . . . . . . . . 106 Rings Hidden Within Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 A Lucky Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Themes and Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 Splitting Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Inside Out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Patterns and Kinship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Duality and Rhombic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Rhombic Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Using Rhombic Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

for further information . . . . . . . . . . . . . . . . 125 index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

about this Book The heart of this book consists of the many projects

The first five chapters present information that is good

you can carve. They are grouped in chapters according

to have before you start carving. However, if you have

to their basic shape—flat rings, twisted rings, knots,

done some carving before, you probably know most of it

and so on. All the projects—including variations and

already. The final chapter presents geometrical knowledge

prototypes for future carving—are shown here to help

for those who really catch on and want to create their

you quickly find the model you are looking for, even

own carving challenges.

if you don’t know its name or type.

I hope you enjoy creating magic woodcarvings as much as I do.

Hopf Link. Page 56.

Lovers’ Hearts. Page 59.

Chains. Page 61.

Borromean Rings. Page 62.

Borromean Variation. Page 63.

Four Triangles. Page 65.

Square Dance Variation.

Page 68.

Six Pentagons. Page 69.

Star Cluster. Page 71.

Clover Leaves. Page 72.

Four Triangles Variation.

Page 66.

Square Dance. Page 67.

Six Pentagons, Variation. Page 70.

Six Pentagons, Holden’s Version. Page 70.

Royal Octalink. Page 73.

Halo. Page 74.

Octoball. Page 76.

Tetraball. Page 77.

Hexaball. Page 77.

Hosohedral Decalink. Page 86.

Hosohedral Hexalink. Page 85.

Double Star. Page 78.

Memento Mori. Page 81.

Flexus. Page 82.

Asalink. Page 84.

Five Inverted Tetrahedra.

Page 87.

Five Tetrahedra. Page 87.

Triple Whitehead. Page 90.

Tetracoil. Page 92.

Wave Packet. Page 94.

Helicon. Page 95.

Hexacoil. Page 96.

Decacoil Prototype. Page 41.

Quintuple Borromeo. Page 97.

Small Tetraknot. Page 99.

Great Tetraknot. Page 100

Inverted Great Tetraknot.

Page 102.

Small Concentric Tetraknot.

Page 103.

Great Concentric Tetraknot.

Page 104.

Knotted Double Star. Page 111.

about the author Many people find most of their identity in work, and may have a preferred activity to enrich their leisure time. A lucky few have managed to combine work with hobby, duty with pleasure.

Magic woodcarving is not the kind of work you do for a living, and yet this is where I find most of my identity. Hence there is little point in dwelling much on what I have done for a living.

It will suffice to mention that a great deal of my life has been occupied with exhibitions, both at museums and at commercial shows. For a few short periods I was also involved with developing companies in the information technology business, and lately I have become involved with special education.

As a woodcarver, I am a complete autodidact. Although I have always enjoyed working with wood, I have never had any formal training. My interest in solid geometry was aroused by a very enthusiastic teacher at the School of Architecture in Copenhagen, where I studied for a couple of years in the sixties.

This is also where I discovered Martin Gardner’s monthly column, “Recreational Mathematics,” in Scientific American, to which I subsequently subscribed for many years. Another major source of inspiration was the Dutch artist M. C. Escher, whose truly magical prints have made a lasting impression on the mathematical and scientific community.

I have had several successful exhibitions that created much attention, but in rather limited audiences. This book will provide me with the largest audience ever.

10

WOODCARVInG Magic

foreword

By GEORGE W . HART chief of content, the museum of mathematics http://momath.org/ http://georgehart.com/

This book is a gift. No matter how much anyone may pay for

One learns best by doing. So I encourage you to take

it, it is a gift from the author to all of us. Bjarne Jespersen’s

up a knife and start the chips flying. You’ll soon come to

immense life-long project is to create original woodcarvings

appreciate fine points of both woodcarving and rigorous

that are beautiful and intellectually rich. I have held many

spatial thought. Beginners will find the ABCs to get started.

of these objects and had the wonderful pleasure of turning

Those with some carving experience will find challenges

them about in my hands, seeing them from many angles, and

to strengthen their technical and spatial visualization skills.

coming to understand their intricate depth. Countless hours of

Experienced carvers will study and appreciate the novel

technical work, creative thought, and repeated refinement are

forms presented. And everyone can savor the beautiful

concentrated into these pages. Whatever royalties may come

photographs of Bjarne’s masterworks.

from book sales, I am sure that Bjarne will be earning practically

As a mathematician, I am tempted to blather on about the

nothing per hour for the lifetime of effort presented here. Enjoy

symmetry groups and the families of polyhedral forms that

his gift to us!

are the foundation for these creations. But I’ll not, as the best

Geometry is a universal in our physical existence. For

way to learn such mathematical ideas is intuitively, by making

example, we live in a three-dimensional world where only

symmetric objects. The power and beauty of the universal,

certain structures are mathematically possible. You can

international language of mathematics will demonstrate itself

assemble six squares into a cube, but you can’t make a closed

along the way.

solid from five or seven squares. It isn’t that you aren’t trying

Bjarne’s journey along this path is nicely presented

hard enough; it is just that space itself, even the empty space

throughout this book. Art and math come together beautifully.

of a vacuum, constrains the possibilities. There is a school of

You will see the power of a focused mind to continually pose

mathematical art that explores the frontiers of what is possible

new challenges. His work moves the field forward and raises

in our space, and Bjarne is a master of this art. He is out to

the bar for other wood carvers. A mere technician might have

uncover the structure of space by looking inside pieces of wood

simply challenged himself to carve longer and longer chains,

and carving away anything that doesn’t look like his visions.

but instead these pages reveal a continuing development of

The technical aspects of fine woodcarving are not my area of expertise, but I have whittled enough to appreciate the

growth in richness of vision.

Behind these entangled loops and links lies a creative human

enormous labor that goes into craftsmanship at Bjarne’s level.

spirit sharing his discoveries and his expertise. In writing a book

It is a shame that with so many distractions in popular culture,

like this, the author knows that many people might walk right

today’s youth might never be exposed to traditional crafts such

past it on the shelf. Some will flip through it quickly and put it

as woodcarving. There is much to be learned when working

down. A few will find inspiration to pick up a knife and start

physically to make something real that otherwise would only

carving. And a special few will embrace these ideas and explore

exist in one’s imagination. It would be fantastic if this book

even further, mastering what they see and spending the time

could start something of a woodcarving movement. These

to push beyond, to make new, original carvings. I hope those

visually engaging models seem so appropriate for today’s high-

explorers remember this gift and find a way to pass on their

tech times.

discoveries as a gift to future generations.

WOODCARVInG Magic

11

introduction We have all seen a magician pull a live pigeon out of an empty hat. We know that is impossible and we wonder how he did it. Similarly, as you can see from the photo, someone has put a wooden pigeon into a wooden cage without breaking the bars.

Again, we know that is impossible and we wonder how he did it.

I have no idea how the magician got the pigeon into the hat—I suspect he had it up his sleeve—but I know how the bird got into the cage. It was in there from the start—both bird and cage were carved from a single block of wood. That is magic woodcarving.

The first magic woodcarving I remember seeing was at the National Museum in Copenhagen. A small barrel was plugged with a wooden peg connected by a wooden chain to a wooden eyebolt, to prevent the peg from being lost. A piece of string could have accomplished the same purpose, so why spend many hours carving a wooden chain— something totally unnatural? Had a string fastened the peg, I would hardly have noticed it. The wooden chain, by contrast, left such an impression that I had to go back to see it again when I moved to Copenhagen several years later. Alas, it was no longer there, but

John Starkey

I still remember it today, more than half a

Magical. A magician may conjure numerous pigeons out of a top hat, but could he conjure this bird into its cage? The magic woodcarver can. John Starkey, of Virginia, is the master behind this carving.

12

WOODCARVInG Magic

lifetime later. It is actually both amusing and satisfying to carve chains and other seemingly impossible objects out of a single piece of wood. The magic lies in the thrilling sensation of seeing the impossible.

Remarkable. An ordinary metal chain hardly merits a second glance.

But a wooden one is another story.

introDuction

Carved rings. The Borromean Rings are symmetrical from every direction, a sort of perfection many other figures lack.

This is very different from the Borromean

some creative geometric techniques I have

Rings. These rings play identical roles,

developed, hoping you may find them useful.

each one surrounding another and passing through the third.

We might say the rings have topological

It is not all geometry, of course. I have included plenty of challenges for readers eager to get started with knife and wood,

symmetry (topology being the study of

anticipating the joyful sensation of cutting

how things are connected—the aspects of

the tiny chip that makes the first part

geometry that do not depend on angles or

come loose.

measurements). The Borromean Rings are also pair-wise separate—only the presence of all three rings links the group properly.

Borromean Rings became the starting point of a long, exciting exploration of the strange universe where I have found most of the models presented in this book.

Simple. Two or three colored links on the chain identify a simple carving project.

It is a universe where subjects like solid geometry, topology, and symmetry reside.

The book may be seen as a travel guide for new visitors to this universe. I will present some fundamental principles and describe

14

WOODCARVInG Magic

Complex. Ten colored links identify an extremely challenging project.

introDuction

The projects are presented in groups of

but failure tends to do the opposite. Do not

different types, and more or less in order

despair if your results look less perfect than

of complexity and difficulty. Each project is

mine. I have been practicing for forty years.

marked with an icon indicating difficulty:

On the other hand, if you make something

Two chain links means easy; ten means very,

you believe will impress me, please let

very difficult.

me know.

The first few projects include step-by-step

Before you start carving, read the

drawings and detailed explanations. However,

chapter on tools and methods. If you are

as you try some of the more difficult projects,

interested in the geometric methods I used

you will need to rely increasingly on your

to develop the models, go to chapter 11,

growing experience. It is just not possible

“Creative Geometry.�

to give detailed instructions for the more

Most people can enjoy my models

complex projects. Do not jump to the

without understanding their mathematical

more complicated projects until you have

background or how to carve them. If you are

successfully completed some of the simpler

that kind of reader, enjoy the pictures and

ones. Success will encourage you to go on,

read whatever you like.

Nested balls. Although intriguing and attractive, nested balls lack the complete symmetry of the Borromean Rings.

WOODCARVInG Magic

15

tr aDitions

The Gotland Museum, Visby.

1

Early magic. A detail of a bridal chair made in the thirteenth century. The loose ring is turned, not carved, but must have seemed magic.

traditions Although I am no expert on the history of

in finding out where and when they were

magic woodcarving, I feel obliged to share

made and to see how far back in time they

what I know about it.

go. In fact, you can find examples from nearly

I have never been able to find an academic study on the subject. There is not even a

The oldest object I am aware of is a

standard term that describes it. “Magic

thirteenth century bridal chair from Källunge

woodcarving” is the name I prefer because I

church on Gothland, now at Gotlands

find the work truly magical.

Museum in Visby. The chair has loose rings

The nature of these carvings denotes a

16

every part of the world.

on some of its turned parts. Although they

community with a certain amount of surplus,

are not hand carved, the rings evoke the same

so anthropologists should have some interest

sort of magic as the carvings.

WOODCARVInG Magic

tr aDitions

Wool winders. Suitors would carve these practical little items to show they were skilled men who expected a diligent mate.

Spoon by DaviD WeStern, picture by chriS robertS.

In rural Europe, it was once common for

of symbols for love and trust, and sometimes

young country workers to carve wool winders

a personal touch relating to the trade or

for their loved ones, with loose balls inside

personality of the carver. They often feature

the handle so they would rattle when used.

loose balls inside the handle, plus links of

A local museum has a couple, with carved

chains at the top. Two spoons may even be

dates of 1799 and 1810. There may be even

connected by a chain—itself a strong symbol.

older examples.

There is a beautiful book about Welsh

In Wales, there is a strong tradition of

lovespoons from the present publisher—Fine

carving lovespoons. They feature a range

Art of Carving Lovespoons by David Western.

Lovespoons. Welsh lovespoons such as these have many features of magic woodcarving, such as nested balls and chain links. Some are even carved as a joined pair of spoons.

This carving tradition has been kept alive. photo courteSy of veSterheim norWegian-american muSeum, Decora, ioWa, uSa.

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17

M o c k- u P s a n D P r o t o t y P e s

4

Mock-ups and Prototypes

Plenty of projects in this book can be carved

mock-ups—three-dimensional models you

by following the drawings and instructions.

can build and modify quickly and easily.

If you can manage most of them, you can

I have used a variety of materials

call yourself a skilled magic woodcarver. But

and techniques to build mock-ups and

if you want to go further, my drawings and

prototypes. The ones I cover here are meant

instructions will not be enough. In order

as an inspiration rather than absolute

to succeed, you will have to build your own

guidelines. You may have access to different

prototypes—carefully crafted, permanent

materials and you may come up with

structures to guide you as you carve. If you go

techniques that I have not thought of. You

even further and start developing your own

may even develop new ideas that require

models, pencil and paper will not be enough

different kinds of models. Your imagination

to bring them to fruition. You will also need

is your most valuable tool. Do not let it be limited by anything you read in this book.

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WOODCARVInG Magic

A useful first step.

A good prototype is useful when carving a complicated model. Above, the prototype and the carving are mirror images, which I do not recommend. In this case, I made the prototype before I learned how the model related to a family of others.

When I carved the model, I changed the orientation to match an earlier carving.

M o c k- u P s a n D P r o t o t y P e s

Hardened clay. Polymer clay is not as easy to work with as ordinary clay, but it can be hardened in the oven. Here, the polymer-clay prototype of a variation of the Borromean Rings is shown on the left. The finished carving is at the right.

clay Two types of clay, readily available at crafts

Polymer clay. This material (Fimo and

or art-supply stores, make good material

Sculpey are the dominant brands) can be

for mock-ups.

hardened in the oven. Polymer clay is not

Modeling clay. It is quick and easy to use,

nearly as easy to work with as modeling clay.

but has one shortcoming: The clay does not

It must be kneaded hard and long before

dry and cannot be fired. So there is no way

you can form it, and the resulting shapes

to get from a clay model to a rigid model.

are rather unstable. Separate parts must be kneaded together at the junction. Still, if you overcome those difficulties the result will be very satisfying.

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43

c age s

The basic octoball. This model has a small solid ball at the center with five shells around it. It is carved in oak with a diameter of about 2 in. (50mm).

Mapping shapes. The three red circles denote the centerlines for the ribbons of the Octoball.

The green and blue arcs define the Hexaball and the Tetraball, respectively.

All of the arcs lie on the angular bisectors between the red circles.

otherwise the inner spheres may not rotate freely. Draw three rings on the surface: one at the equator and two (perpendicular to each other) from pole to pole. Cut away

Difficulty rating: 5

the triangular sections between the rings, leaving three mutually perpendicular flat disks. Actually, you may prefer to narrow the disks toward the center for a better balance between rings and holes in the smaller spheres.

Carve outward from the center. Begin the carving with the small ball at the center.

If you work from the outside in, you will have trouble keeping the parts fixed while you work, and may damage the outer shells while working on the inner ones. Thin shells like these are rather delicate, so make the outer shell slightly thicker than the others.

The cubic and tetrahedral variations shown at right are just two alternative designs.

You can produce both from the six circles obtained by halving the right angles between the three circles of the first model.

Two variations. These two variations might be called Hexaball and Tetraball. Hexaball, carved in rosewood, has a diameter of 1 7/8 in. (48mm). Tetraball, carved in olive, is about 1 3/4 in. (45mm) in diameter.

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77

c age s

A stellar shape.

The Double Star is such a marvel I can hardly believe I discovered it myself. But I did, and I carved it in pear. It measures about 1 3/4 in.

(45mm) on a side.

Double star Difficulty rating: 8

The concept of linked cages arose from

that engage each other symmetrically. This

an attempt to combine the best qualities

wonderful Double Star, my first endeavor,

of ring models and nested spheres. Ring

was also my best.

models have intricate overall symmetry even though individual parts are extremely

this model. Here, I will show how to carve it.

simple structures. Nested spheres may have

Begin with a cube. Draw two diagonals

very symmetrical parts, but they exhibit no

on each face. Cut along the diagonal lines

mutual symmetry.

from the corners down to the centers of the

I wanted a model in which each part has a high degree of symmetry and with parts

78

On page 110, I explain how I developed

WOODCARVInG Magic

faces. Use a fine-toothed saw and cut just outside the lines, leaving a little material

knotteD rings

10

knotted rings The rings forming these models twist so

them particularly beautiful. Larger numbers

much they tie themselves in knots. I call

of knots are possible. This chapter features

models like these polyknots to differentiate

three tetraknots I have carved, as well as

them from polylinks, the generally accepted

prototypes for two other promising models.

term for linked rings or polygons.

I have concentrated on tetraknots (models with four knots) because I find some of

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WOODCARVInG Magic

Knotted Rings

Interesting interior. Notice how the axes of fourfold rotation symmetry meet at right angles, and how each axis is surrounded by four twisting curves—one from each knot.

This gives the model a very interesting interior.

W oodcarvi n g M a g i c

105

c r e at i v e g e o M e t r y

Tracing the family. Now, find out what the new family looks like. Begin by coloring a single part on each solid. Then find a well-shaped three-dimensional representation for each family member. You can do that with a pipe-cleaners mock-up.

at the rotation axes (i.e. the vertices of the

application like Adobe Illustrator to create

rhombus)? How many parts of the model

the pattern. That way, you can also draw

cross each face of the pyramid? Mark these

in the outline of the rhombic pattern,

crossings as points on the edges of the

squeeze it to fit, and place it on all the faces.

rhombus. How are these crossing points

(Always draw the pattern as an outline.)

connected? Do the parts cross one another

The alternative is to draw the rhombic

within the rhombus or at its edges? The

pattern by hand and paste photocopies

actual shape of the parts is not important; a

of it on the faces of the pattern.

schematic representation is better because it will be used in five different models.

Constructing mock-ups for a given pattern

mock-ups, you will probably be eager to study the structures each one represents.

is straightforward, at least at first. Begin

The best way to do that is to fill in their

by making models of the three rhombic

individual parts with colored markers.

solids from cardboard or stiff paper. The

122

After cutting, folding, and gluing the

If you are satisfied with the result, move

illustrations on the facing page show the

on to more realistic models. Pipe cleaners

basic shapes, including tabs you fold and

are ideal for this purpose, but you need care,

glue to hold the mock-up together. I highly

patience, concentration, and dexterity, too.

recommend using a computer drawing

Chapter 4 gives you some useful hints.

WOODCARVInG Magic

c r e at i v e g e o M e t r y

Rhombic Dodecahedron

Cube (Hexahedron)

Rhombic Triacontahedron

Cut and paste. Once you have a rhombic pattern, you can apply it to the rhombic solids in a very literal way. Prepare the shapes shown here, cut them out, and fold them into the three different solids. It is best if you can generate the shapes on a computer, so you can easily stretch the pattern to fit the rhombs in both directions and copy it to all faces.

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123

Membership

c r e at i v e g e o M e t r y

Family

6

12h

12v

30h

30v

a6 Borromean rings, 62; a12h, v four triangles, 65; a30h, v six Pentagons, 69.

A

A6

A12h

A12v

A30h

A30v

B6 four triangles, 65; B12h, v square Dance, 67; B30h, v star cluster, 71.

B

C

B6

C6

B12h

C12h

B12v

C12v

B30h

C30h

B30v

C30v

c6 triple Whitehead, 90; c12h tetracoil, 92; c12v Wave Packet, 94; c30h hexacoil, 95; c30v halo, 74.

D6 cloverleaves (holden’s version), 72.

D

D6

D12h

D12v

D30h

D30v

e6 helicon, 95; e12h royal octalink (holden’s version), 73.

E

E6

E12h

E12v

E30h

E30v

f6 inverted great tetraknot, 102.

F

F6

F12h

F12v

F30h

F30v

family table When I first tried to arrange my models in a family table, I discovered several unexpected relationships. The table also had some empty slots, awaiting new discoveries. Some of my best models, such as Hexacoil and Wave Packet (pages 96 and 94), filled some of those slots. Others revealed interesting results that were not suitable for carving, either because they only had a single part or because the parts would not support each other but collapse into a mess. Some of the slots only held repetitions or mirror versions of other models.

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WOODCARVInG Magic

Family table. This sample chart gives you some idea of how I used rhombic patterns to organize models into families.

The process led to new discoveries, as with the Royal Family in row three, of which I only knew two when I started.

Maybe you will come up with a pattern that produces an equally interesting family. If you are curious about which of these discoveries made it into the book, that information can be found at the end of each row.

f o r f u r t h e r i n f o r M at i o n

for further information Books, monographs, and Internet sources for more about carving, geometry, and the connection between the two .

Books

Monographs and articles

•

•

Holden, Alan, Orderly Tangles: Cloverleafs, Gordian Knots,

and Regular Polylinks, Columbia University Press, 1983 .

Hart, George W .: Orderly Tangles Revisited . Proceedings

of Bridges 2005 (http://bridgesmathart.org). Hart explores

A pioneering work on linked structures in which I found

creative and artistic use of Alan Holden’s work and presents

several of my own discoveries as well as many others .

relevant custom-made software .

•

Holden, Alan, Shapes, Space, and Symmetry, Columbia

University Press, 1971 . A good starting point and source

•

Hull, Thomas (ed .), Origami3, A K Peters Ltd ., 2002

(with 15 page article by Robert J . Lang: Polypolyhedra

of inspiration for anyone who wants to work with regular

in Origami, a slightly abridged version of the online

shapes in 3D space .

article mentioned below .

•

Springett, David, Woodturning Wizardry, revised edition, Fox Chapel Publishing Company, 2007 . The book includes excellent instruction on how to produce nested spheres and other marvels on the lathe .

•

Tangerman, E . J ., Whittling and Woodcarving, Whittlesey

•

Jespersen, Bjarne: Geometric Tools for the Magic Woodcarver .

Proceedings of Bridges 2010, p .151, (http://bridgesmathart.org) .

Contains part of Chapter 11 of this book .

•

Jespersen, Bjarne: Three Approaches to Regular Linked Structures. Proceedings of Bridges 2011, p .139 .

House and McGraw-Hill, 1936, Dover edition, 1962 .

(http://bridgesmathart.org) . Compares my own work

With 36 pages dedicated to chains, balls in cages,

to that of Alan Holden and Robert Lang .

and other whimsies .

•

Tibetts, Malcolm . The Art of Segmented Woodturning:

A Step-by-Step Guide. Linden Publishing, 2006 . Contains models similar to some of mine, beautifully crafted as segmented turnings .

•

Wenninger, Magnus J ., Polyhedron Models, Cambridge University Press, 1970 . The author presents cardboard models of interesting polyhedra with instructions on how to make them .

•

Western, David, Fine Art of Carving Lovespoons, Fox Chapel

online resources

•

Hart, George W .: Encyclopedia of Polyhedra . A comprehensive online reference that includes an historic overview of polyhedra in art and an annotated bibliography . Part of George Hart’s impressive web site, www.georgehart.com .

• •

Jespersen, Bjarne: Pocket Art . Contains a selection of magic

woodcarvings at: www.lommekunst.dk/gallery.html .

Lang, Robert J .: Origami Burrs and Polypolyhedra, Part III, at www.langorigami.com/science/polypolyhedra/Polypolyhedra_

Publishing Company, 2008 . An excellent guide to the history

part_3.pdf . Lang defines a class of regular linked structures

of lovespoons from Wales and elsewhere, with patterns and

and proves that exactly 54 such structures exist . He

instruction for many of the author’s designs and guidelines

rediscovered all of Holden’s regular polylinks and several

for making your own .

of my models .

•

Lang, Robert J .: Renderings of the 54 polypolyhedra in dowels at: www.langorigami.com/science/polypolyhedra/

polypolyhedra-dowels/polypolyhedra-dowels.php4.

WOODCARVInG Magic

125

about Bjarne jespersen Bjarne Jespersen is an internationally known woodcarver and sculptor with a passion for mathematics and geometry. He uses creative geometry to give new life to “magic woodcarving”—carving a piece of wood into sections that are loose but cannot be separated. He resides in Denmark.

craFTS & HoBBiES/Woodworking

$19.95

There is a school of mathematical art that explores the frontiers of what is possible in a given space, and author Bjarne jespersen is a master of this art. In Woodcarving Magic, he will show you how to uncover the structure of space by looking inside pieces of wood and carving away anything that doesn’t match your vision. With more than 25 designs for carving “tricky objects” with connected parts that are loose, yet cannot be taken apart, you’ll learn to craft astonishing linked cages, knotted rings, and more.

You’ll learn how to get started with advice on choosing wood, selecting tools, and preparing blanks. Woodcarving Magic has projects for both the beginner and the experienced carver, featuring step-by-step diagrams and detailed instructions

seemingly

that break down the process into achievable steps. Those with

impossible

objects from a

single piece of wood

engineering minds will enjoy the author’s explanations of his creative geometric methods. Those with mathematical minds will marvel at how beautifully art and math come together. and nd those who enjoy woodcarving will find challenges to strengthen their skills and will appreciate the novel forms presented.

“This beautiful and extraordinary book not only teaches, but inspires and awes.

Reconciling the impossible and possible has always been my aspiration, but I've also realized it's necessary to do so with an established harmony of thought, emotion, and human touch. The organic material of wood yields itself naturally to this challenge; I used wood to create the first prototype of the Cube.

Original rubik’s Cube® used by permission of seven Towns Ltd. www.rubiks.com

To succeed at such tasks, ordinary or extravagant, brings pure joy—as I am convinced readers of this book will testify.”

– Ernö Rubik, creator of the rubik's cube