Table of Contents

PHI and the Golden Mean as it applies to dressage... and do you know how far you have ridden in USA Training Level One 2003? I double dare you to do it on foot (walk, trot, canter) before your ride your horse! Very aerobic! And an eye-opener for what we ask of our horses!

Information About the Book
Ordering Information:
Caveat to the reader: This Atlas has opinions based on data, a Bibliograpy (selected annotated books) and a Literature Cited composed of journal articles that themselves have useful citations. If the concepts in the Atlas send you on a quest for more information, then the book
has served its purpose. Those ideas may or may not be yours, they are just a place to start shedding light upon many expositions about classical equitation, and recent departures from its teachings. In other words, "When all candles bee out, all cats be gray."
John Heywood (1497?–1580?), Proverbes. Part i. Chap. v.

Table of Contents
(Biomechanical Riding & Dressage: A Rider's Atlas)
Copyright ©2006 Dr. Nancy Nicholson
All rights reserved.
All materials contained in this publication are protected by United States copyright law and may not be reproduced, distributed, transmitted, displayed, published or broadcast without the prior written permission of Dr. Nancy Nicholson.
ISBN 0-9778102-1-6
Zip Publishing
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Anatomy and Function

Chapter One.
A Gymnastic Curriculum for Partners

•Dressage is in the Details: Honing Perception and Sensation
•Biomechanical Riding Summary: Visualization and Sensation
•Learning Styles and Strategies
•Growing With a “Training Tree”

Chapter Five.
Transitions Between Gaits
•Transitions: The Needle(s) in the Haystack
•Walk to Trot, Trot to Walk
•Trot to Canter, Canter to Trot
•Walk to Canter, Canter to Walk
•Transitions on Free Longe
•Transitions in Dressage Tests: Table of Percentages
•Transitions and “On the Forehand” and “In Horizontal Balance”
•Transitions Within Gaits (Moderate Collection and Extension)
Chapter Two.
Anatomy and the Independent Seat

•Ladies and Gentlemen on Horseback
•Anatomy of the Independent Seat
•Exercises for an Independent Seat
•The Seat and Rider Position: Effects on the Horse
•The Seat and Rider Position: Problems & Solutions
•Independent Seat and Riding Forward Into Halt
•Diagonal Riding and Muscles of the Horse’s Trunk
•Biomechanical Riding Spiral Seat: How It Works
•Saddling: Between Horse and Rider
•Saddling and Cinchiness
Chapter Six.
Lateral Suppleness and Range of Motion

•Lateral Bending and Other Movements of Horse’s Spine
•Summary of Amount of Bend for Lateral Work
•Leg Yield
•Shoulder-in, Shoulder-fore
•Half Pass, Haunches-in
•Turn on the Haunches, Pirouette (Walk)
Chapter Three.
Anatomy and “On the Aids”

•On the Contact, On the Bit, Behind the Vertical
•Contact, Throughness and Preparation for Transitions Within Gaits
•Long and Low (Anatomy of Aids for Work in Deep Posture)
•Work in Deep Posture and Its Effects on Breathing
•Lateral Balance: Straightness
•Horizontal Balance: Relative Elevation, Self-Carriage and
Special Topics and Maintaining a Trained Horse

Chapter Seven: Fine Tuning Balance and Coordination
•Canter and Pirouettes
•Canter and Flying Changes of Leg
•Rein-Back and Shaukel
•Collected Walk, School Walk, Passage & Piaffe Transitions

Chapter Four.
Anatomy and Function: Dressage Gaits

•Horses and Dressage Gaits: Between Ground and Rider
•Dressage Gaits: Swing and Ground Contact Time
•Symmetrical and Asymmetrical Gaits
•Ordinary Dressage Gaits: Walk, Trot, Canter
•Canter (Again) and Disunited Canter
•Staccato and Legato: the Music of the Gaits
•Assessing Dressage Gaits with Digital Video Records
Chapter Eight.
Mind and Matter: Sustaining a Gymnastic Foundation

•What Was Not In This Atlas (and Xenophon’s advice...)
•Resistances and Evasions
•Maintaining a Trained or Older Dressage Horse
•Selected Problems:
Lateral Walk, Balancé, Goat-on-a-Mountain Piaffe,
Tongue Out, Magpie Hopping (flying changes of leg)


The animations and still frames in this book are based on selected data from literatures on biomechanics, mathematics, robotics, veterinary medicine/anatomy and dressage. Each field of study has its own conventions about marking gait cycles and priorities for analysis. Studies of locomotion with different objectives produce information that can be usefully integrated, provided there is a method for convenient comparison between disciplines.

1) In order to be consistent in presenting gait cycles, it was necessary to adopt a single convention selected from three available for the initial/start moment of the stride cycle of walk, trot or canter.

I chose the LEFT HIND TOE DOWN moment to initiate a stride cycle. This is the convention of the biomechanics literature and it allowed matching cycle phases in all three fields of study. You will see this instant indicated as a five pointed star in appropriate images.

The mathematics literature looks at left fore toe-down as the inital point of a gait cycle and it ignores the problem of representing asymmetric gaits like canter and gallop, considering the pattern (repeated template) to be the same in principle, albeit reversed for the right or left half of an asymmetric gait. Because canter, gallop and cross canter are asymmetric gaits, the left hind toe-down convention from biomechanics for the left canter would require that the stride be shown with the diagonal pair as the initial condition (left hind-right fore), an unnecessary complication.

The dressage literature designates one gait as having an initial condition: the outside hind of the canter landing out of suspension. It does not specify where the symmetrical gait cycles of walk and trot should begin. You see right canter in my animations and in all my diagrams: the left hind toe-down is the start point for the stride. I use the right canter and right gallop and left cross canter as representatives of those gaits.

The benefit of this unified convention is that the image showing a horse at a consistently chosen instant of a gait is a "snapshot" of the phase (relative limb position) characteristics of the gait allowing useful timing comparisons among data sets. These comparisons of limb positions are crucial for understanding transitions, the most frequenly ridden movement in dressage tests. DRESSAGE transitions between gaits are a fraction of the possible transitions among gaits and are dependent on deliberate tempo (the slower part of the tempo range). Correct dressage transitions (fluent, level, prompt) are also foundation skills for gymnastic training of dressage horses.

2) Every effort has been made to be anatomically accurate with images. All images other than digital photographs are not drawings, but are output from a computer model based on veterinary dissection manuals and radiographs, checked against digital video frames. And I do know that there are two sesamoids per leg and that rib cages have two sides! The technical advantage of a computer model is that its motions represent proportions of the body unchanged from one image to the next in a sequence. Drawings, no matter how talented the artist, represent what an artist thinks is ocurring, not what may actually happen during movement. Among the critical exercises for this are the "long and low" and bending issues: drawings are not to be trusted under these technical circumstances. I am aware that horses have two sesamoids on each fetlock and two halves of a rib cage: I have left out extra detail where I wish to emphasize a feature of gait or of anatomy. This decision to simplify also cuts down digital file size. Screen resolution with scaled down representations of horses and riders is another limit on accuracy as well as on precision. Finally, the orbit of the top of the scapula is not yet known with great accuracy for each gait, nor has the warping of the rib cage (felt by riders during gait) been quantified.
3) I trained my horses to work at liberty for the video sequences used to check the computer model. I felt it was useful to have substantial material presented without influence from the presence of a rider or of tack. Further, at liberty the horses could indicate their understanding of a movement or exercise by their body positions or expressions. This turned out to be crucial for understanding transitions because managing the velocity and phase differences between gaits are important neurologic skills. Horses in this book represent two types of warmblood and two types of "baroque" conformation, with differing "typical" body proportions.
4) My horses also provided important feedback for my own riding skills. Humility is a handy thing to have around intelligent large animals... and in this Age of Litigation I can say whatever I please about my own mistakes without being sued.
5) I have busted 4 5 6 Macintosh computers with this project... there is no such thing as too much memory, storage space or processor speed. Some of the more complex animations have 8000+ objects to follow. My recurring nightmare is that somewhere in cyberspace there is a little lost patella... and a pelvis that is occasionally elusive. The computer models are originally vector graphics rasterized (computed shapes converted to picture elements or "pixels") only at the end of an export to movie format or for incorporation in the Atlas. This retains the shapes and proportions of bones, a crucial feature of a model's correspondence to real-world riding issues.

Whoever said digital publishing was simple was on another planet... or in another Universe...

TOP Covers for the spiral bound copy (bound along left edge, plastic cover, metal spiral), 8.5" x 11" paper.

Sample from Introduction to Chapter One: A Gymnastic Curriculum for Partners

TOPSample images:
Copyright ©2006 Dr. Nancy Nicholson
All rights reserved.

Selected Problems Series:
Goat-on-a-Mountain Piaffe
(Ch 8)
Gait Analysis Series (Ch. 4)
Goat on a Mountain Piaffe
Anatomy Series (End Poster)
Rider Position Series (Ch. 2)
Lateral Work Series (Ch. 6)
Lateral Work Series (Ch. 6)
Lateral Work Series (Ch. 6)
Lateral Work Series (Ch. 6)
Transition Series (Ch. 5)
Canter to Trot (with half halts explained)
Transition Series (Ch. 5)
Trot to Canter on the Forehand
Transition Series (Ch. 5)
Canter to Walk (with half halts explained)
Transition Series (Ch. 5)
Canter to Walk, Free Longe
Special Topics Series (Ch. 7)
Mounted and Unmounted Canter Pirouettes
Maintaining a Trained Horse (Ch. 8)
Asking Horses What They Know on Free Longe...

TOPFrom star systems to planets to organisms to micro-organisms, to molecules and electron shells within atoms, special ratios represent fundamental relationships between the basic elements of form in the architecture of Nature. Those ratios are proportions generated from rules in the mathematical world that happen to match patterns in the geometries of life. It is these geometries that frame many physical interactions between the bodies of horse and rider. They are unified by the proportion of the Golden Mean.

If you ask "Why should patterns from mathematics match patterns in the real world?" I will answer that pattern matching is an early phenomenon of human mental activity. It is a potent way to learn. Further, using it with care allows us to focus on major aspects of a topic and to see a basic form. Where form/pattern in nature departs from that in math, we are attracted to important areas of investigation. To reprase a line from a Gilbert and Sullivan's operetta, "Where doth pattern lie? In the mind or in the eye? Where is pattern bred? In the heart or in the head..?"

Pi is the best known of these preferred proportions: the relationship between a line and a circle. Pi is defined as the ratio of the circumference to its diameter. Numerically, Pi nearly equals 3.1415... Question, if you ride the circumference of a 20 meter diameter circle in a dressage test, how far have you ridden? Answer: distance ridden (Pi x diameter) = 3.1415 x 20 meters = 62.8319 meters.

USA Training Level One 2003
Distance Ridden, 20 m x 60 m arena
20 m Circles (4 x 62.83 m) trot and canter
= 251.32
diagonal in free walk (63.25 m)
= 63.25
2.5 times around the arena: w, t, c segments
= 400.00
2 center lines (60 m)
= 120.00
834.57 meters!:
a kilometer is 1000 meters!
or 0.52 miles!

Less well known is e, the numeric base for natural logrithms. While our number system is based on place values of 10, Nature's exponential geometry is based on the e ratio, which approximately equals 2.76... .

Forgotten amidst the modern rejection of math's mystic meanings is PHI, a natural ratio almost equal to 1.610339...

Pi, e and PHI are not ordinary finite numbers with an exact value. Rather, these relationships don't have a precise numerical specification, either as a decimal or a fraction. This imprecise uncertainty is expressed by the three dots following the last digit to indicate the number is a non-repeating decimal. Math labels these unspecifiable ratios "irrational," yet Nature apparently makes profuse use of these ratios in multitudes of forms — from star systems to viruses and electron orbitals. However, there is a caveat: not all phenomena take the ratio as fundamental. For instance, the ordering of seeds in sunflower heads and bracts in pine cones is close to the Fibonacci sequence, but the development of sunflower leaves does not follow that ordering.

The ratio of the Fibonacci sequence of numbers, made notorious by Dan Brown's DaVinci Code (which is a series of cipher puzzles and not a code, except for a Biblical reference) converges on PHI. For instance, you can construct your own Fibonacci sequence in the following fashion:

Generate the numbers The series is now: The Ratio of the numbers converge on PHI, or 1.6180339...
start with 0 & 1 1 1/1 = 1
0, 1 1 2/1 = 2
1 + 1 2 3/2 = 1.5
1 + 2 3 5/3 = 1.66666666667
2 + 3 5 8/5 = 1.6
3 + 5 8 13/8 = 1.625
8 + 5 13 21/13 = 1.61538461538
13 + 8 21 34/21 = 1.61904761905
21 + 13 34... and so forth...

Here is my "Vitruvian Horse" based on the Fibonacci Spiral, Golden Mean, the Golden Spiral... drawn in green (I liked the color) against the Golden Rectangle (white lines). Leonardo DaVinci trained himself to draw with the Golden Mean ratio, which is one reason his work is so magnificently credible, regardless of subject. His plants and animals show the ratio in their arrangement of parts. The Fibonacci numbers, which you can find by the recursive formula a(n) = a(n-1) + a(n-2) with [a(1) = 1, a(2) = 1, n > 2] gives

TRIVIA (below)

Hypnerotomachia Poliphili:
Francesco Colonna
the bronze Equus infelicitatis image from page 32
One of the celebratory processions from page 171. Most of the quadruped figures in the book,
in this case unicorns, are shown in the walk in this phase of the limbs, as though the author thought
it a particularly aesthetic moment... The author also created some early animation sequences,
brought to "life" on the MIT site for the book.
Wikipedia has an entry for this tome...

Facts do not cease to exist because they are ignored.
Aldous Huxley, Proper Studies