2024 Workshop Weekend with Handell and Halbert Composition Workshop

 



Nov 10, DAY 2 Workshop with Karen Halbert WORKSHOP NOTES

Weekend with Handell and Halbert
A Deep Dive into Rocks, Water & Dynamic Symmetry: Dual Workshops with Albert Handell and Karen Halbert
November 9 and 10, 2024, 9 a.m. to 4:30 p.m. at Bluebird Studios, Santa Fe

A Right-Brain Approach to Painting and a Left-Brain Approach to Composition Students may participate in either or both sessions.

All images copyright.

OUTLINE

Foreword
Karen Halbert - Radical Impressionist: A Mathematician Paints Introduction
Videos
Basic Dynamic Symmetry Armature Construction
Paintings with Basic Dynamic Symmetry Armature
Dynamic Symmetry Tools and Field Example
Expanded Dynamic Symmetry Armature Construction (advanced) Paintings with Expanded Dynamic Symmetry Armature
Golden Rectangles
Root Rectangles
Demo-Turbulent Lands Anatomy
Additional Painting and Armature Examples (Extra)
General Appendix:
Select Bibliography 



Foreword

I would like to thank Albert Handell, Artist Extraordinaire and long-time mentor. Albert encouraged me in my study of Dynamic Symmetry years ago in one of the first mentoring sessions I had with him. Albert suggested that the topic, Dynamic Symmetry, was a topic that would attract other artists. Since that time, I have adopted the brand name, Radical Impressionist: A Mathematician Paints, the name of my blog, http://www.karenhalbert.blogspot.com/, with many posts on Dynamic Symmetry. And now, I begin my second year of giving workshops on the topic. I am including a few examples of Albert’s “intuitive” paintings for analysis with Dynamic Symmetry armatures.

A few years later, during the pandemic, I took zoom workshops with Michele Byrne, primarily to learn how she paints figures with a palette knife. Much to my delight, her workshops included an in-depth study of Dynamic Symmetry, with examples and links to talks she’s given on the topic. I was in my element. She suggested my name to be included in an article on Dynamic Symmetry in the Plein Air Magazine, titled “The Science of Beauty”. You may read it here for now (may be removed): https://online.flippingbook.com/view/354061990/.

While putting together my revised class notes for this second workshop, I discovered that one of my favorite painters, Douglas Fryer, has been teaching composition and Dynamic Symmetry for years in Universities and in his videos and workshops. I have included a discussion of an armature he has used in these notes. I am grateful for his expertise which helped me simplify some of the aspects of Dynamic Symmetry.

A few of last year’s workshop students have been using the Dynamic Symmetry armatures in their paintings this past year – among them, Carole Belliveau and Cynthia Inson and Beth Cooper. We have had extensive email exchanges on the topic, which have aided me immensely. Carole has followed along with me, sharing thoughts on Fryer. Also, Carole had introduced me to Coy Ludwig, author of the noted book on Maxfield Parrish. Cynthia introduced me to the artist, Sean Michael Chavez, an artist creatively incorporating armature lines in his paintings (not Dynamic Symmetry, but worthy of study).

And thank you to my other students, who helped me find ways to explain the mathematics behind Dynamic Symmetry.

I have taken many workshops with Bill Gallen -in Santa Fe and at Ghost Ranch- and have learned much about color theory and composition from him. I was delighted to see that he incorporates the use of golden calipers when he designs a painting, along with a zen-like approach in the composition stage to identify what the painting is about.

I would also like to thank my close artist friends, Judy Howells and Ruth Hogan, for their feedback and for their welcome company on painting expeditions.

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Karen Halbert - Radical Impressionist: A Mathematician Paints

Karen Halbert, former college professor of Mathematics and current Landscape Painter, presents the armature systems from an artist's point of view. Her background in Mathematics gives her a unique perspective, however. In this workshop you will get a glimpse of how a Mathematician perceives the underlying framework of paintings.

Visit https://karenhalbert.blogspot.com/2022/03/a-mathematician-paints-article-resumed.html for a little more insight into Halbert’s journey to this point in time. Below: view from my office before I made the decision to retire and paint – but after the construction of a new tower.












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Introduction 

Overview:

From the Classical Greek Era to the present, artists have been using armatures or grid systems based on musical and mathematical principles to help create harmonious paintings.

In this workshop, Dynamic Symmetry is explained and compared with the Intuitive approach described in Handell’s book, Intuitive Composition: A Right Brain Approach to achieving simplicity, harmony and balance in your paintings written in 1988.

Armatures

An armature is a grid applied to a painting support to be used to aid in the design of a painting. Why use an armature when designing a composition? Most professionals begin with a design, whether it's an architectural blueprint or a house plan. What does a house scaffolding look like? It's a mesh of triangles, squares or diamonds, a network that is as strong or sturdy as needed. Could the painter benefit from such a scaffolding? Or is a scaffolding too restrictive? Can a firm foundation be used creatively? I have heard excellent, creative artists say that they now rely upon an armature.

Perhaps the experienced artist uses an armature intuitively? We have all learned the rule of thirds (ROT), which itself is an armature. How is dynamic symmetry different? The ROT guides the artist toward not making a completely symmetric (perhaps boring) composition. The Dynamic Symmetry Armature does more than this, though, one could argue. It adds excitement to the composition. It provides directional lines and several points of interest distributed across the canvas that might trigger the artist's awareness of potential places to paint elements to be noticed. The armatures might help the artist plan how to direct the viewer's eye around the canvas.

Can the experienced artist benefit from the Dynamic Symmetry Armature? Let’s see the advantages of the maze of lines in the armature, if any. Why are they dynamic or harmonic? This will be discussed in the workshop.

An additional benefit is that armatures may be used to analyze the design of paintings after their completion. Dynamic Symmetry and other Armatures will be used to analyze the composition in this workshop in paintings by Albert Handell, Karen Halbert, Douglas Fryer, Michele Byrne and others. Did their work benefit from the use of these armatures implicitly or explicitly? Is the very nature of armatures such that it’s natural for artists to arrange their compositions to align with the Dynamic Symmetry -or other- armatures?

An article on Dynamic Symmetry by Bob Bahr was published in Oct 2024 titled: The Science of Beauty. The article contains painting images and quotes by Douglas Fryer, Michele Byrne and Karen Halbert and others.

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The systems to be referenced in this workshop include the

  • Rule of Thirds - Basic Dynamic Symmetry 4 to 3 Armature and Rule of Thirds

  • Harmonic Armature - Harmonic Armature Construction in the Appendix and Thomas 

    Kegler

  • Basic Dynamic Symmetry - Basic Dynamic Symmetry Armature Construction

  • Expanded Dynamic Symmetry - Expanded Dynamic Symmetry Armature Construction 

    (advanced)

  • Informal Subdivision (See Andrew Loomis Informal Subdivision Example) 

though the major emphasis will be Dynamic Symmetry Armatures.

Demonstrations may include a look at tools for students to use, such as Photo Shop, the iPhone and iPad Wise Photo Apps (WISE Photo App) and the Snapseed App for phone. Here are the icons on my phone:



A demonstration is included in these notes, showing a process for utilizing an armature to design and construct a painting: Demo-Turbulent Lands Anatomy. Basically, the artist may begin with a photo or a scene and superimpose an armature (or view it through an armature transparency), moving the photo until a focal point lines up with the desired armature focal point for example. Lines leading up to the focal point can be aligned with the armature lines. Then the artist might sketch a value study to try to line up the photo (with armature). Then the grid could be drawn on the panel and the painting proceeds from there.

Adding an Armature to a Photo

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Intuitive Composition

In the forward to his book Intuitive Composition, Albert Handell references Jay Hambidge's The Elements of Dynamic Symmetry, praising Hambidge for his knowledge and inspiration. However, Handell acknowledges a significant hurdle for aspiring young artists: the impracticality of incorporating Dynamic Symmetry into their practice. This workshop will delve into how the principles of Dynamic Symmetry can be made accessible and practical while exploring the deeper meanings behind intuitive composition. By examining the interplay of harmony, simplicity, and shapes, we can uncover the common threads that bind these concepts together.

The concept of “seeing” is central to Handell's approach. “Mentally the artist begins to draw, paint and compose intuitively all the time.” Developing this intuitive vision is not merely a natural gift; it is a skill that can be cultivated through practice and heightened perception. As Handell notes, the longer one paints, the more one begins to see in terms of rhythms and flows.

The exploration of intuitive composition through the lens of Dynamic Symmetry reveals a wealth of possibilities for artists seeking to enhance their practice. By embracing the principles of harmony, simplicity, and the effective use of shapes, artists can develop a deeper understanding of their work. The workshop will serve as a platform for participants to experiment with these concepts, utilizing armatures to create compositions that resonate on multiple levels. Through this exploration, we may find that there is indeed more in common between these seemingly disparate approaches than initially meets the eye.

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Videos

See the Bibliography for recommended reading and videos.

You can ”google” golden spiral or symmetry and get many different, interesting, sometimes “fun” online links.
(new) One youtube video that I recently found: https://www.youtube.com/watch?v=c8ccsE_IumM. Golden Ratio: MindBlown! By DavidsonArtOnline is worth watching (9 minutes).

(Thank you, Carole, for the first two recommendations here.)

Basic Dynamic Symmetry Armature Construction

Now, for the instructions on constructing a Basic Dynamic Symmetry Armature. We will review a few paintings with Basic Dynamic Symmetry Armature Overlays before actually doing the construction, however. (We’ll come back to this page and do the steps one at a time.)
One can use marks on the top and bottom edges for the endpoints of the reciprocals rather than construct this each time. See the 12x16 Basic Dynamic Symmetry Panel Measurement in the Appendix for one example. For other panel measurements see: Basic Dynamic Symmetry Panel Measurements also in the Appendix. These measurements can be used to check your own construction measurements.

Main Diagonals plus one ‘reciprocal’.



Draw diagonal lines from corner to corner. The one from the lower left to the upper right corner is called the Baroque Diagonal; it is considered the more pleasing since we actually read from left to right and usually from bottom to top. The diagonal from the top left to the bottom right is called the Sinister Diagonal; it creates more tension. The two diagonals form an “X” dividing your rectangular panel (in half horizontally and vertically). Drop a perpendicular line (known as a reciprocal) from the lower right corner through the Baroque diagonal at 90 degrees, ending on the top edge. Use the 90 degree corner of another panel, a 90 degree triangle or a T-square to line up this reciprocal so that it forms a 90 degree angle with the main diagonal.

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Main Diagonals with all four Reciprocals.


Drop three additional lines perpendicular to the main diagonals from the corners. Can you identify the other three intersections of main diagonals and their reciprocals?

(Final) Basic Dynamic Symmetry Armature

This is the basic Dynamic Symmetry Armature. It has the main diagonals and their
reciprocals plus 4 main verticals and horizontals through the intersections of the diagonals and their reciprocals. Let's call these intersection points the "eyes" of the armature.

4:3 Basic Dynamic Symmetry Armature

This seems to be a sturdy scaffolding. The 4 eyes are at points of maximum thrust since the angle between the two intersecting lines is maximized at 90 degrees. The eyes also have several lines radiating from them, potential directional lines to draw the viewers' eye around the canvas. These could be potential focal points.

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Panel Proportion Example



Comparison of the angle for “Baroque” Diagonals for panels with the same proportion: 4 to 3: 8/6 = 12/9 = 16/12 = 4/3. With Aspect Ratio, 1.333...
For these panels with the same aspect ratio or proportions, the diagonals should line up as indicated in this photo.

You can see how easy it is to be off by a little when measuring the right 90 degree angle formed by the Baroque Diagonal and its reciprocal. (see erasure marks). I checked measurements in the appendix before committing too much,

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Basic Dynamic Symmetry 4 to 3 Armature and Rule of Thirds

Armatures we have all used include the Rule of Thirds; this example provides a comparison of the 4 to 3 Basic Dynamic Symmetry Grid with the Rule of Thirds for general reference and discussion. See also Comparison of Dynamic Symmetry (basic) Armature with Rule of Thirds Additional Examples in the appendix.



4 to 3 Dynamic Symmetry Compared with the Rule of Thirds

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Intuitive Composition and Dynamic Symmetry

To be explored is how an artist might be guided by the fundamental concepts of Dynamic Symmetry along with Intuitive Composition design elements from Handell’s book:

  • Placement of points of interest near intersections of the lines, often providing maximum thrust and movement.

  • Placement of verticals and horizontals at irregular intervals leading to a better balance (that might align with the armature),

  • Directional lines to lead the eye toward main points of interest (diagonal lines including the main diagonals).

  • Areas marked by a variety of pleasing and harmonious geometric shapes. I like the “kite” shapes that are generated by the Expanded Dynamic Symmetry Armature.

  • Horizontal lines through points of intersection might help in the placement of the

    horizon line or other lines in our landscape such as the tops or bottoms of cliffs or waterfalls. 

 Paintings with

Basic Dynamic Symmetry Armature

Paintings with Basic Dynamic Symmetry Armatures will be viewed first before looking into more expanded armatures. It’s of interest to see how the Armature aligns with paintings or how the paintings “lock into” the armature (think focal points or areas or how the cliff edges or waterfalls parallel the grid lines or actually follow them), whether painted intuitively or guided by a grid. For example, does Handell intuitively use divisions that follow Dynamic Symmetry or is it just that the Dynamic Symmetry armatures are naturally divided and spaced harmoniously, well balanced and encouraging different shapes? And dividing the space in this manner is intuitive to the experienced artist. Note in these two cases the difference in proportions and how the grids cross each other at different positions at the top (and bottom).

Albert Handell


Albert Handell, Quarry Falls, 12x16, pastel Basic Dynamic Signature Armature

Albert Handell, Waterfall Demo, 16x20, Oil

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Karen Halbert

Karen Halbert, Turbulent Waters III, 9x12, oil After Albert Handel
Basic Dynamic Symmetry Armature

The focal area was chosen to be the area in the lower right around the intersection of the main sinister diagonal (upper left corner down to lower right corner) and its upper right reciprocal, the perpendicular line from the upper right corner through this diagonal.
Note the line of the waterfall leading to this focal area.

The top trio of trees in the center seems to point to a central focal area. In fact, perhaps the main part of the waterfall as a whole is the focal area- in the lower center, but with different non- symmetric elements to the left and right. Or perhaps if the waterfall were moved down to the right then there would be no ambiguity as to the focal point. And it would be at the location of the most dynamic thrust or play between the main diagonal and its reciprocal (a future painting). An emphasis on the sinister diagonal versus the 'baroque' diagonal is said to lead to a more active or turbulent artwork.

Beth Cooper


Beth Cooper, Courtyard Queen, 12x9, Basic Dynamic Symmetry

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Carole Belliveau


Carole Belliveau, Fly Fishing 3, 8x6, BASIC Dynamic Symmetry Armature

Note how the Fisherman’s head is at one of the desired focal points, an intersection of a main diagonal and its reciprocal. And the vertical through this point follows the body and its reflection. And the pole goes through one of the other desired focal points. Carole is an intuitive painter with many years of experience, winning awards and showing in galleries. Again, an example of the meeting of intuition and logic.

Carole has been adding the grids to her colorful panels to take into the field, ready for the next painting.

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Carole Belliveau, Photo Reference and Painting Start 16x20, with some Basic DS Pencil marks

Basic Dynamic Symmetry Armature Overlay on Painting Start

Carole Belliveau, Los Alamos Landscape, 16x20, oil.

From Carole: “Here is what I am working on now- a studio painting 16x20. I am including the original photo and the basic start on the painting with the lines of the grid showing through. Note that the original shot was a larger view and that I moved the transparent grid around to find the best alignments, cropped my reference accordingly and then drew the grid on the panel.” .... Finished painting.

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Cynthia Inson


 



Cynthia Inson, 9x12 Cloudscape, Oil. Stages

Cynthia will take a scenic photograph and do paintings of various proportions based on the scene.

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She has been experimenting with different armatures.
A little teaser with a Harmonic Armature Example:
See Harmonic Armature Construction on this post for a little more information. The Root 1 and the Root 4 Rectangles sometimes benefit from a Harmonic Armature, perhaps since they are wider. Note that this 10x20 is a Root Rectangle (Root 4) painting. We will go over Root Rectangles soon.


Cynthia Inson, 10x20 Beam Me Up Painting Start with visible pencil marks.


Cynthia chose a harmonic armature for this 10x20 painting. 

Dynamic Symmetry Tools and Field Example WISE Photo App

Perhaps it's time to consider using the iPhone's app - to look more closely at the next examples with the Wise Photo app. I've experimented with the app at length and it's great fun but the app works best with either a golden rectangle (eg, 10x16, 5x8, 8x13, 13x21, 21x34, 15x24), for the Spiral overlay or a camera proportion such as 4:3, 3:2 or 16:9 proportion for the BASIC Dynamic Symmetry Grid. Other proportions will be cropped off. Here's a link to order it:

https://apps.apple.com/us/app/wise-photos/id1406085029 (you will have to use this link from your phone since the app is not available on a computer; the first two rules are free from apple; it costs a few dollars otherwise) for the iPhone and iPad. I’ve been informed that future versions will have enhancements such as a Harmonic Armature choice.
And here are images created by the app:






Albert Handell, Taos, 18x24, oil. ROT and Basic Dynamic Symmetry

 There are many choices for grids including “Vanishing Point”:


Note how the spiral fits part of this 3 to 4 proportioned painting only (appropriately since the spiral has a proportion closer to 1 to 1.618 rather than 1 to 1.333...)

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WISE Photo App Sample Grid Descriptions






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Field Example

When I am in the field, I often take a photograph of the scene I wish to paint. Then I open the WISE Photo App on my iPhone and look at various grids applied to the image, but especially the Dynamic Symmetry grid. I always have a Dynamic Symmetry grid drawn on my panel (in pencil) so I can then line up my panel next to my phone with the grid on the image.
Here’s just a recent example of a painting done up at Ghost Ranch:




The app allows one to download the image with the grid overlay.

This example illustrates also how one can take a photograph of the painting itself and then apply the WISE App to obtain the grid overlay as above (to check against the original PHOTO overlay).

I try to line up the image cliff edges or focal points and hill lines to align as much as possible with the grid lines or intersection points. Another big step is to identify the horizon line (and try to use one of the horizontal lines in the grid; in the case of the basic dynamic symmetry there are only the two main horizontals. But we can draw a horizontal through the “X” intersection at the bottom, here above the Dynamic Symmetry label). In this case, I actually used a horizontal line from the Expanded grid for the horizon line at the bottom of the cliff (more later on the expanded grid).

I found images of the initial stages of the painting (I happened to use an “expanded” grid here):





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Armature Transparencies



I also use an armature transparency to look at paintings and the reference photo or scene while I create the painting and after the painting is completed to check.

Link to blog post on how to create a transparency (I will hand out some sample transparencies):

https://karenhalbert.blogspot.com/2023/07/constructing-transparency-viewfinder.html.



These are generally 5x7 including the darker frame. The openings could be made small enough that you can view any photo or painting with the transparency on top of the phone’s camera viewer. However, the transparency images for the class will be generally larger to fit within the mat’s 4x6 opening.

But we’ll take a break now for the students to construct their own grids on panels.......

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Expanded Dynamic Symmetry Armature Construction (advanced)

Now to study Expanded Dynamic Symmetry Armatures:

There are two main parts of the expanded dynamic symmetry armature that provide a mathematical basis to be used when designing one’s composition:

1) Self-similar rectangles with the same proportion. Dynamic Symmetry seems to go hand in hand with what we see in nature. One of the design principles of dynamic symmetry is the concept of shapes that repeat themselves in smaller and smaller (or larger and larger) imprints - self-similar rectangles. This division into self-similar rectangles of the same proportion is at the heart of Dynamic Symmetry, illustrating the magical ‘self- similarity’ so prized by the Classical Greeks and by Renaissance artists.

2) Unit Square. the square shape was seen in antiquity as a fundamental unit in design. We will study the “rebatement” of squares and will also analyze the square as “dynamic”.

These concepts provide a sturdy but pleasing scaffolding with areas to be selected as main focus areas for a painting, with additional lines and curves leading to these areas.

Note that the additional diagonals resulting from the expanded construction can create added excitement, movement and dynamism for a painting.

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Dynamic Symmetry Self-Similar Rectangles Diagram



  1. Blue main diagonal lines from corner to corner.

  2. Turquoise diagonal lines* (known as reciprocals) from the corners perpendicular (90

    degree) to the main diagonals extending to the top and bottom edges (form the diagonals of the self-similar, smaller vertical rectangles; see the Self-Similar Rectangle Theorem in the separate Mathematics Appendix.

  3. Vertical and Horizontal gray lines through the intersections of the main diagonals with the reciprocals.

  4. Turquoise vertical lines from the top and bottom edges where the reciprocals meet the edges.

Note that 1) – 3) describe the Basic Dynamic Symmetry Armature. Adding the lines in 4) result in the self-similar rectangles.

The eyes -or potential focal points- are at points of maximum thrust since the angle between the two intersecting lines is maximized at 90 degrees. The “eyes” also have several lines radiating from them, potential directional lines to draw the viewers' eye around the canvas

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Dynamic Symmetry Rebated Squares Diagram


Right Rebated Square

  1. Pink verticals mark off squares with sides equal to the shorter side of the main panel; the panel sides are “rebated” to form a square.
  2. Pink diagonals of the squares are included in the diagram
  3. Do the same to the left side for the full “rebated square diagram”
One more aspect of a fully expanded Dynamic Symmetry Armature is rebatement of the squares (also known as rabatment). Rebatement of the squares is based on the theory that a square is a fundamental unit and as such it is appealing to viewers; a painting with perceived squares or derivations of squares is calming and natural. Classical artists began to use rebatement of the squares to design their paintings. Its use continues to this day.

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Final Expanded Dynamic Symmetry Armature Diagram

Proportional, self-similar rectangles (turquoise).
Rebated Squares (pink).

A shortcut to constructing the full diagram is to mark off the intersections on the top (and bottom) edges. The table for the Basic Dynamic Symmetry Panel Measurements in the Appendix may be used for the points defining the self-similar rectangles. For the pink rebated squares, just mark off the height on the top (for the left rebated square) and the width minus the height (for the right rebated square).

The additional diagonals drawn from the corners to the reciprocal intercepts on the edge help the artist find elements in the painting to help lead the eye around the painting, adding excitement and movement.

The Expanded Dynamic Symmetry Armature allows the artist to take advantage of several fundamental elements of dynamic symmetry: rebated squares emphasize how the composition has a fundamental unit (square) as a basis; self-similar rectangles build on the importance of repeating shapes at different magnifications – something comfortable to our eyes.

Tavis Leaf Glover has a youtube video with some instructions on constructing a Dynamic Symmetry Grid for canvas sizes: https://www.youtube.com/watch?v=PaUsB57UF1Y (reviewing all of our terms such as reciprocals and root rectangles).


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Paintings with Expanded Dynamic Symmetry Armature

Albert Handell



Albert Handell, Rocks and Water, 12x16. EXPANDED Dynamic Symmetry Armature



Albert Handell At the Interior Falls, 18x24, EXPANDED Dynamic Symmetry Armature

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Albert Handell, Elusive, 16x20, oil, Expanded Dynamic Symmetry Armature



Albert Handell, Morning Light, 12x16, 1st Place, Pastel, 26th Annual Masterworks of NM EXPANDED Dynamic Symmetry Armature

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Michele Byrne


Michele Byrne, Parc Restaurant, 16”x12” - Cobra oil paint EXPANDED Dynamic Symmetry Armature

Michele is known for her use of the Dynamic Symmetry Armature – an expanded one to include the Self-Similar Rectangles and Rebated Squares. Her Facebook posts frequently include steps where you can see pencil marks from the armature in the intermediate steps of her painting.

Michele has been told by her followers that her compositions improved after she began using the Expanded Dynamic Symmetry Armature; she started winning even more awards.

She has a very informative article that I highly recommend, available through the Oil Painters of America: https://www.oilpaintersofamerica.com/2022/03/dynamic-symmetry-and-how-i-incorporate-it-into-my-plein-air-and-studio-and-practice/

And a video presentation by Michele on using dynamic symmetry in art, Using Dynamic Symmetry in Art

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Karen Halbert

Karen Halbert, Ghost Ranch Cabin, 9x12, Judge’s Special Recognition Award, Paint NM! Expanded Dynamic Symmetry Armature



Karen Halbert, Turbulent Waters, 12x16, oil with Expanded Dynamic Symmetry Armature

I consciously used a Dynamic Symmetry Armature when I designed this painting. My intention for the main focal point was the upper left desirable one, the intersection between the Sinister Diagonal and its reciprocal from the lower left corner. I missed it by a bit. But note how the top edge of the waterfall is parallel with the main (uplifting) Baroque Diagonal from the lower left corner to the upper right corner.

From my website description: “Fond memories of many walks along rivers and creeks seeing the turbulence in the waters and listening to the sounds. I like the 's-shape', zig-zag of the river/waterfall to help draw the viewer's eye around the painting. The ripples of the water also keep the viewer's eye moving. The brightness of the waterfall makes it a natural focal point (area). “

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Evelyne Boren



Evelyn Boren, Lavender Harvest Los Poblanos, 30x36, Watercolor Expanded Dynamic Symmetry Armature

Evelyne’s very dynamic paintings work well with the Expanded Dynamic Symmetry Armature perhaps due to the use of many directional lines (diagonals) in her paintings.
Of interest is how several verticals and diagonals align with the slopes of the roofs or touch the buildings at peaks. Also, the first lavender rows seem to parallel the diagonals (of the rebated square or of the full panel). I like the way two of the figures zig-zag up the middle, leading to the main structure. And there’s a nice curve leading up the hill as well.



Informal Subdivision

This is an Andrew Loomis Informal Subdivision Grid overlaid on her painting. See Andrew Loomis Informal Subdivision Example in the Appendix for more information.

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Golden Rectangles Introduction

Jay Hambidge coined the term “Dynamic Rectangle” to include “golden” and “root” rectangles, shapes with special properties that incorporate the unit square and self-similar rectangles in such a way that they promote many harmonious combinations of these basic units.

The canvas size that is arguably the most pleasing is the golden spiral. Examples of panel sizes that approximate the Golden Rectangle are: 10x16, 5x8, 8x13, 13x21, 21x34, ... These are the panel proportions that I have used for my paintings for many years. I’ve added the golden spiral to the basic dynamic symmetry armature I’ve constructed to be used for “golden rectangles”.

Note that for the Golden Rectangle the division of the space is special: it consists of a square plus a smaller vertical rectangle with the same proportions as the original golden rectangle. This is in fact what makes this shape so special; no other proportion leads to this result.


PHI Basic Dynamic Symmetry plus Golden Spiral

More information on the Golden Spiral is available in the appendix in the Golden and Whirling Squares Spiral section, including the Fibonacci Series.

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Golden Mean Calipers with Bill Gallen:

Well known artist, Bill Gallen, uses a golden caliper in the field to help ensure unequal, natural measurements between important elements- while he sets up the painting elements. The ratio of the two indicated distances is 1:1.618..., the golden ratio. I ordered my caliper from https://goldenmeancalipers.com/ (delivery - two weeks from New Zealand)



Plastic surgeons use the caliper for their measurements; measurements of body parts divide the space naturally in a golden ratio.

Example Caliper Usage from 2018:



Karen Halbert, Beginning Design with DS Armature and Caliper (See addendum at end for links to making your own)

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Paintings with the Golden Spiral Armature Albert Handell



Albert Handell, Rocky Falls, 24x40, Near Golden Rectangle Armature plus Light Spiral

Karen Halbert


Karen Halbert, Turbulent Lands 15x25 3:5 AR 1.67 2024 Masterworks 3rd Place, Oil/Acrylics
Dynamic Symmetry Armature with the Golden Spiral

Turbulent Lands was painted in the studio, based on several plein air works in the past couple of

years. Included in these notes is a detailed demonstration of this painting (Demo-Turbulent Lands Anatomy).

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WISE Photo App Golden Triangle Armature

A simplified armature for this proportion is called the Golden Triangle. Here’s an example from the WISE Photo App with a description:


Handell Taos Winter Golden Triangle Armature




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Root Rectangles Introduction

Root Rectangles are also considered pleasing (along with golden rectangles). Root rectangles lie at the heart of Dynamic Symmetry. Root Rectangle have special properties, which make their Dynamic Symmetry Armatures unique. It’s best that we use canvas sizes approximating the Root Rectangle proportions. However, in general, popular panel sizes are not Root Rectangles.

From Wikipedia: A root rectangle is a rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as √2, √3, etc.[2]

Root Rectangle Construction

Here’s an ingenious way to construct rectangles with lengths Root 1, Root 2, ... Root 5. (Hint: Pythagorean Theorem: a squared + b squared = hypotenuse squared).



By Jay Hambidge - Dynamic symmetry: the Greek vase, Public Domain,

https://commons.wikimedia.org/w/index.php?curid=4119863

The Root 2 rectangle, for example, has sides Root 2 (see bottom edge from A to C) and the vertical 1. The Root 3 rectangle, is formed from the diagonal of the Root 2 Rectangle, “arcing” it down to E on the line. The distance from A to E then is Root 3.

“Root” Painting Panel Sizes

A few common panel sizes are explicitly “Root” rectangles: square (Root 1) and 1:2 panels (Root 4); eg, 10x20. Other not common sizes include Root 3; eg, 10x17.32 and Root 5, 10 x 22.36.

  • Root 3 = 1.732.. can be approximated by: for example, 7x12 (since 12/7 = 1.714..).

  • Root 5 = 2.236.. can be approximated by: for example, 6x14 (since 14/6 = 2.33..).

  • A Root PHI canvas is close to an 11x14 panel since Root PHI is approximately 1.272, while

    14/11 = 1.272727...

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Square (Root 1) Armatures and Paintings Douglas Fryer

Recently I had the fortune to review again the Composition portion of Douglas Fryer’s masterful and beautifully detailed video through Streamline. During the Composition Stage, he explained each step in composing his painting. He used Photoshop to manipulate a photo of a farm snapped out the window as he was driving by. These steps included the initial plans for the proportion of the painting and how he moved elements around to more pleasing locations. He then showed each step in constructing an armature over the photoshopped image that would be used as a guide in the painting: a grid that he has designed, one that utilizes the full depth of squares, root rectangles, circles and arcs, all fundamental shapes pleasing to the eye. (The intersections of the quarter circle arcs with the main diagonals mark off the Root 2 rectangle.)

Doug Fryer, Farmhouse Photoshopped Reference with the “Fryer” Square Armature

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Fryer Square Armature* used by Fryer for his Square Farmhouse painting on the Video

Fryer uses different composition armatures, selecting them to fit the painting. He might choose Root 2 subdivisions within the square as in the Farmhouse painting, or other Dynamic Divisions.

The intersections of the Root N rectangle diagonals with the arc locate the Root N+1 Root Rectangles; eg, the diagonal of the Root 2 Rectangle (with the orange top) intersects the arc at the endpoint of the diagonal of a Root 3 Rectangle.

See Root Rectangles inscribed in a Square (advanced) in the Appendix for a detailed description of constructing root rectangles within a square as was done here.

He also likes to show simplified harmonic (rational) subdivisions, with verticals at the thirds, quarters, halves or even fifths as well as horizontals. Subdivisions at fifths was a popular device

used in historical paintings.
*As constructed by this author based on the steps in Fryer’s video (without the red triangles).

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Doug Fryer, Farmhouse Video Painting with the “Fryer” Armature

(Note that the “grid” colors change when superimposed on the colored areas of the painting, even though the opacity was at 100%.)

Fryer’s exploration of horizon lines and geometric divisions exemplifies his understanding of visual stability and harmony. Fryer recognizes the psychological impact of these compositional choices; the horizon suggests calmness, while deliberate spatial divisions can convey a sense of order and balance. He asserts, “Composition is really what moves us,” highlighting that the arrangement of marks is not a mere technical endeavor but a vital emotional catalyst.

His understanding of the land as shape, pattern, and edges enables him to create a visual language that resonates deeply with viewers.

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Natasha Isenhour


Natasha Isenhour, Metronome, 20x20, oil on canvas with the Fryer Square Armature

Natasha is an intuitive painter. Recently I chatted with her about her process - to be discussed further in the class. Her painting, Metronome, is an excellent example of how intuition and logic meet. Moreover, it is amazing that without being aware of the “Fryer” armature, her painting aligns with it so well. Natasha commented that the vertical subject in a square canvas intrigued her.

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Albert Handell Armatures:

Albert Handell, Deep Woods, 28x28 Oil Painters of America, 2024 National Juried Salon Silver Award


Two Root 4 Dynamic Symmetry Armatures 

Harmonic Armature



“Fryer” Square Armature

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Michele Byrne

Michele is known for her use of the Expanded Dynamic Symmetry Armature. With a square format, she uses a double armature consisting of two Root 4 (1:2) rectangular grids:



Michele Byrne uses a “Thrust Map” sketched on a 4x6 index card to get started. In this case, she seems to have placed the dominant horizontal at the level of the seated figures. The dominant vertical is drawn through the main waiter. And a dominant diagonal along a shadow, which seems to be near the main sinister diagonal of the bottom 1:2 rectangle if it were shown.




Michele Byrne, Afternoon Delight, 12x12, oil

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Other Root Rectangle Paintings

Karen Halbert
Diablo Canyon 8x16, oil. Root 4 1:2 Armatures:



Basic Dynamic Symmetry Armature

Expanded Dynamic Symmetry Armature Phase 1


4 Root 4 Side by Side Dynamic Symmetry Armature



Harmonic Armature

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Albert Handell



Albert Handell, On the Rio Grande, 14x24, oil, Root 3 Expanded Dynamic Symmetry Armature Image from Handell’s original Intuitive Composition.

Douglas Fryer



Douglas Fryer, Midsummer Greens, 12x28, oil. Root 5 Armature
Fryer also superimposed a simplified grid on this painting to show how the artist can reduce the armature and still have some help in laying out the main areas of the painting:



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George Bellows Dynamic Symmetry Application

From Jay Hambidge, Lectures, 1923:

https://archive.org/details/dynamicsymmetryi0000jayh/page/10/mode/1up?view=theater

Example of Bellows’ application of Dynamic Symmetry:

“From subsequent talks with him [Bellows] and after I had seen some of his sketches and paintings, wherein he had employed dynamic symmetry, I was struck by the logic of his thinking and his grasp of the essential ideas involved in pattern planning.”

“Last year Mr. Bellows was awarded first prize at the International Art Exhibition in Pittsburgh for his painting “Eleanor, Jean and Anna.” Later this picture was bought by the Rochester, New York, Museum for a price of $7,500.00. He has given me two of the sketches he made for the picture and a description of his method in planning and carrying out the dynamic composition.”






“The shape of the canvas was half of a root five rectangle. The half of a rectangle is composed of two similar figures to the whole. In this case one figure is over the other. ‘The subdivisions of the major area were made by the application of squares on either end, reciprocals and their diagonals and an arrangement of the major shape into its two squares and an extreme and mean ratio rectangle. ‘The scheme could hardly be simpler and Mr. Bellows’ sketch of his plan furnishes an excellent illustration of the inherent harmonies of areas which the dynamic rectangles contain.

The ratio of the canvas, end to side, is 1: 1.118, or 1: 2.236 divided by 2. The central horizontal line dividing the two root five rectangles (AC and QR) passes through the child’s mouth and her head, above this line, occupies the square WZ. The area XC is a square and XQ is composed of two extreme and mean ratio rectangles. These areas, of course, are duplicated in the root five rectangle, AC, above.

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The interesting and to modern artistic practice revolutionary feature of Mr. Bellows’ picture arrangement now appears. The making of the sketches and their assemblage in the picture involve different rectangles. The method by which this is brought about would be utterly confusing to an artist of the old school who depended upon simple squares for enlargement and reduction. Mr. Bellows has taken advantage of the most distinguishing property of the dynamic rectangle, the almost unlimited possibility of subdivision or addition in simple recognizable terms which are inherent in it. In this case the artist picked out a theme and adhered to it strictly. Countless other themes could be selected. Dynamic Symmetry is a system of notation in areas and, like the natural notation of numbers, a series or a scheme adaptable to any particular purpose may be selected. The advantage of this particular area notation lies in the fact that it introduces asymmetric balance and thus avoids the dead commonplaceness of equal units of area, a curse of pictorial composition for ages.”

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Demo-Turbulent Lands Anatomy

Visit the more detailed description on my blog post:

https://karenhalbert.blogspot.com/2024/05/anatomy-of-painting-turbulent-lands.html

I've been painting in the Los Alamos area for a few years. The cliffs leading up to this famous town are striking. I became obsessed with trying to capture their feeling - the shapes and texture and the light on them at different times of the day.
I visited the scene many times, took photographs and placed an armature on one photo, trying to absorb the landscape intuitively. I then superimposed an armature on a photo. I had a 15x25 panel (close enough to a golden proportion, 1.666... rather than 1.618...) and decided to do a painting I titled Turbulent Lands.

I then did a value study on a 4x6 card and looked at a b/w reference. .

I used different techniques even beyond utilizing a golden spiral expanded dynamic symmetry armature in the design in this painting. In particular, I used tools such as putty knives, squeegee and wedge as well as brushes and palette knives. I find that the straight edge tools work very well with an armature. I drag paint across the panel from the line intersections or edges of the armature. I allow the scraping to remove or pick up paint from previous layers, hoping to recreate the texture of the hills and cliffs.

Note that I decided at this point that the upper right focal point seemed to be a better main focal point, leaving the upper left polar point as a secondary focal point as seen in the diagram at the end of this post (and the beginning). The main hill leads up to the right along the Baroque Diagonal passing through the eye of the Spiral, the new intended location of the focus of the painting. The secondary focus is across the painting at the upper left polar point.

I felt this version was not finished so I continued working on it, smoothing out some of the roughness. I brightened some cliffs and added some accents and declared it done:

I submitted it to the 2024 New Mexico Masterworks show and it won a good prize: Third Place in Oils/Acrylics!!


Karen Halbert, Turbulent Lands, 15x25, oil Third Place, 2024 NM Masterworks Oil/acrylics Studio Painting



Karen Halbert, Turbulent Lands, 15x25, Golden Spiral plus Dynamic Symmetry Armature

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Additional Painting and Armature Examples (Extra)

Albert Handell

Albert Handell, Elusive, 16x20, 4:5 BASIC Dynamic Symmetry Armature



Albert Handell, The Rocks at Kaaterskill, 24x30, 4:5 Expanded Dynamic Symmetry Armature From Intuitive Composition (Triangular Movement)

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ARMATURES: Albert Handell, At the Katterskill Falls, 22x22


Two Root 4 Dynamic Symmetry Armature 


Harmonic Armature


“Fryer” Square Armature

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Handell Paintings from Intuitive Composition



Albert Handell, The Stream, 16x22, 3:4 Expanded Dynamic Symmetry Armature (AR 1.375)

Albert Handell, Upper Canyon Road, 30x36 5:6 Basic Dynamic Symmetry Armature (AR 1.2)

Albert Handell, The White Pine, 36x30, 6:5 Basic Dynamic Symmetry Armature

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Douglas Fryer

Can you see the hidden geometry in some of Fryer’s other paintings? I don’t know which armatures Doug used for these specific paintings, but I thought it might be constructive to superimpose the square armature from his video.



Fryer, Final Cutting with “Fryer” Square Armature 



Fryer, Passing Rain with “Fryer” Square Armature

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And of course, Doug has paintings with different proportions. I have superimposed armatures that I might have used for this size. One might try to see where the verticals and horizontals might line up on the painting, if drawn through the armature intersections.



Doug Fryer, Stone Home, 10x20, oil. Basic Root 4 Dynamic Symmetry Armature

Stone Home with Dynamic Symmetry Side by Side Root 4 Armature

Douglas likes to use Root Rectangle proportions: here we’ve seen Squares (Root 1) and Root 4 rectangles, 1 to 2 rectangles.

See more examples of Douglas’ work here: https://douglasfryer.blogspot.com/

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Thomas Kegler

Kegler uses a Harmonic Armature for his paintings. Kegler has provided composition resources with descriptions of the Harmonic Armature. Visit the Resources’ section of his website here: https://thomaskegler.com/resources/
(scroll down and then select Process followed by Composition Tips or Designing a Composition).



You may see works by Kegler in Santa Fe at the Santa Fe Trails Fine Arts on Water Street.

It can be shown that the divisions formed from horizontals and verticals through the intersections of the 14 lines of the armature divide the canvas into halves, quarters, thirds, fifths and sixths, nice rational numbers so desired by the ancient Greeks.

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Carole Belliveau



Carole Belliveau, Morning in Galisteo, 8x10, oil with Basic Dynamic Symmetry Armature

Comments from Carole:
“I rarely use the 8x10 format because I think of it as being shortened on the wide
side for landscape. I have always gone to the 9x12. However, as I became more experienced I saw that each format is a challenge. 8x10 forces you to come up close and personal to a subject and in the case of this particular view I had to eliminate whole sections of the view on the left side that were not all that important and generalize shapes on that side. The painting was done in 30 minutes or so because it started to drizzle and we left. It wasn't until I got home that I realized that it was "cooked and done"! ...

I just got home from a very discouraging plein air session with a less than stellar painting. Driving home I was thinking I should hang up my painting knives permanently when I received this totally unexpected message:”

“Congratulations! Your painting, "Overcast in Galesteo", was an award winner in the July 2024 BoldBrush Painting competition. A huge heartfelt thanks goes out to the amazing painter Ramona Younquist for her encouragement! I believe I will paint another day!”

Note: I believe that Carole did not use the indicated Dynamic Symmetry Armature but I love how it seems to align nicely with the armature.

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Michele Byrne



Square Double Root 4 Armature 



“Fryer” Square Armature



Michele Byrne, Cathedral from an old-fashioned Photograph, 48x48.oil

Michele used the first dynamic symmetry armature, a double Root 4, within the Root 1. As an experiment I applied the “Fryer” armature also.

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Douglas Fryer provided examples from Cezanne and Vermeer with Dynamic Symmetry analysis:

Cezanne: Card Players



Cezanne, Card Players, Root 2 Proportion

Vermeer: Woman Reading a Letter



Vermeer, Woman Reading a Letter, with 4 to 3 proportion Dynamic Symmetry Armature

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Maxfield Parrish Daybreak

I am not presenting an example from Maxfield Parrish in this second class. However. I would like to include a note from the important book on Parrish by Cory Ludwig: on page 128, Ludwig quotes from Parrish, “Of late years I have been doing all my work on a layout, so to speak, of Dynamic Symmetry, a rediscovery on the part of Jay Hambidge of the old Greek method of making rectangles. Those familiar with this method consider that these dynamic rectangles are far more pleasing than just arbitrary rectangle, and not only that, but by their sub-divisions, they permit every feature of the picture to be a harmonious part of the whole panel. If done with intelligence there is a certain balance to the design which is of great value...
“Now these dynamic rectangles are very simple things, but they are fixed and you can have only certain ones, or compounds ones. – This may sound like sheer lunacy but take my word for it, there is a lot in it...

Parrish is one of the most famous artists using the Armature and writing about the use.



Daybreak by Maxfield Parrish with truer colors PLUS expanded Dynamic Symmetry Armature

Parrish seemed to enjoy communicating his methods with the public.
Historically, it is said that some master painters kept the use of armatures as a secret. In many cases, drawings of the armature cannot be found easily.

Wikipedia: " While Parrish's techniques may have been tedious, almost every single fantasy illustrator you can name borrows from the technique and style of Maxfield Parrish, ensuring that his visions live on.”

A more complete analysis of this painting may be found in Scott McDonald’s article: http://www.scottmcd.net/artanalysis/?p=53:

The article also gives an excellent account of Root Rectangles and the Golden Rectangle that might be of interest.

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General Appendix:
12x16 Basic Dynamic Symmetry Panel Measurement

  1. For the following panel, mark the top and boUom edges with points at the indicated inch marks. (7 and 9 top and boUom).

  2. Draw diagonal lines (reciprocals) perpendicular to the main diagonals to these marks. As indicated. There will be 4 reciprocals in all. These are the marks made by the endpoints of the reciprocals to the main diagonals. (These mark off division points that can be used for self-similar ver2cal rectangles in the Expanded Armature; proof of this propor2on is in the full workshop notes. Note that 16/12 = 12/9).

  3. For the Basic Dynamic Symmetry grid, construct verticals and horizontals through the sweet spots with potential focal points: the intersections of the main diagonals with their reciprocals.



Do the same for other panel sizes and proportions, calculating the marks at the top (and bottom) so that if verticals were dropped from these points, then the vertical rectangles would have the same proportion as the main canvas (in this case: 9/12 = 12/16). For a 9x12, the marks would be at: 5.25 and 6.25 (9/6.75 = 12/9=1.3333...)

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Basic Dynamic Symmetry Panel Measurements





Formulas for other panels:

The formula for the Top Intercept: Baroque Diagonal Reciprocal is:

width – (height * height)/width

The formula for the Top Intercept: Sinister Diagonal Reciprocal is:

(height*height)/width

Check that these two intercepts add up to the width. Also note, for example that for the 6x8 panel the proportion of the implied vertical rectangle is 4.5 x 6 the same as 6 x 8. Note that the intercepts switch positions on the top edge for the last three (wider) panel sizes; ie, the Baroque Reciprocal is moved to the right of the Sinister Reciprocal intercept

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Comparison of Dynamic Symmetry (basic) Armature with Rule of Thirds Additional Examples






5 to 4 and 4 to 3 Dynamic Symmetry Compared with the Rule of Thirds






Root 2 and PHI Dynamic Symmetry Compared with Rule of Thirds


Root 4 Dynamic Symmetry compared with Rule of Thirds

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Harmonic Armature Construction



  1. Construct the main diagonals from corner to corner, forming a big X.

  2. Pencil in dotted verticals and horizontals through the center point, the intersection of the two main diagonals. The edges are divided in two by the vertical and horizontal lines at

    edge midpoints.

  3. Draw lines from the midpoints to the opposite corners (forming “flat” Xs).

  4. Draw lines from the midpoints to the midpoints of adjacent edges (forming Xs in the

    quarters).

  5. Count the number of lines. Do you get 14?

Proponents of the harmonic armature refer to the “musical” divisions created by the lines; they divide the canvas in thirds, fourths, fifths and sixths. See the Appendix: Harmonic Armature Proportions and an Extension of Rule of Thirds.

Some might argue that the Harmonic Armature lacks the underlying structure that is a critical element of Dynamic Symmetry, but it does have its own harmony.

Note that for the harmonic armature, ‘stretching’ does not affect the proportion of the distances between the intersection points. The Dynamic Symmetry Armatures are more complex to construct since the reciprocals MUST be perpendicular to the main diagonals; this is a crucial part of the theory behind Dynamic Symmetry.

The artist, Thomas Kegler, has much information about the Harmonic Armature here in his section: Thomas Kegler.

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Andrew Loomis Informal Subdivision Example

I wanted to apply an Informal Subdivision Armature to one of my paintings. I had written a blog post on the Informal Subdivision method written by Andrew Loomis in his “bible of illustration”, Creative Illustration. Refer to this post, https://karenhalbert.blogspot.com/2022/10/creative-illustration-andrew-loomis.html with a more detailed description of Informal Subdivision.

For the Informal Subdivision Grid, I began with 3 lines (Diagonal, Vertical and Horizontal
(through a Golden Point on the upper right since I knew I wanted this to be the focal point) but
one can begin anywhere, with a main diagonal first. Then I chose an additional intersection point
almost arbitrarily for the next triad set of lines and repeated this process as described in the
Loomis book from the 1920s: Creative Illustration. Constructing these lines is a very creative
process - and enjoyable as well. I tried to color code each triad, but sometimes the lines
overlap. I did consciously try to choose intersection points that would result in lines leading to
the Golden Point. (Aside: Loomis specifically recommends NOT putting the focal point on a
“Main” diagonal, which for me contradicts my choice of this golden point for a focal point. But
for the full design, we would make sure not to “fall off” the canvas at the corners by inserting
elements that would in fact stop the viewer’s eye along the diagonal at some point).
Kevin MacPherson utilizes the Loomis’ Informal Subdivision Method in his Magic Grid.

This is an older painting that was designed using a golden spiral with the upper right polar point on the top of the waterfall as the focal point. To analyze the use of informal subdivision here, I constructed the armature here according to the Andrew Loomis guidelines, but intentionally chose the initial horizontal/vertical/diagonal triad to match the “golden point” on the upper right. If in fact this had been the grid on my canvas, I might have placed some of the rocks and trees in better alignment with the grid.







Santa Fe River Turbulence, 10x16, oil. PAAC 2016 Award Informal Subdivision Grid
by Karen Halbert

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I also constructed the informal subdivision grid constructed by Andrew Loomis in his book and have applied it here. I “flipped” it to get more activity around the top of the waterfall.

But since in fact I had the golden spiral in mind when I painted this, I am including the painting overlaid with the Dynamic Symmetry Armature plus a Golden Spiral. I intentionally placed rocks following the spiral curve and other lines leading to the upper right focal point to draw the viewer’s eye around and up to the waterfall.



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Golden and Whirling Squares Spiral

Much of this information in this section is taken from Wikipedia.


Whirling Squares (with 'eye' at lower right)

Construction of the Whirling Squares' Spiral

The light grey squares are marked off within this rectangle. They form what are called the whirling squares. Note that in a golden rectangle with aspect ratio 1.618... a smaller golden rectangle is formed with the part left over after constructing the square. And this continues as we 'whirl' around.

https://en.wikipedia.org/wiki/Fibonacci_sequence:
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1. Starting from 0 and 1, the first few values in the sequence are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...

The DNA molecule, the program for all life, is based on the golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.




34 and 21 are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates PHI, the golden mean, 1.6180339...

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I like to use Fibonacci numbers as lengths for my canvas; eg, 5x8", 8x13" and multiples of the 5x8: 10x16" and 15x24" (near-PHI paintings, where PHI is the golden mean - 1.618...). I prefer these proportioned canvases.

There are many excellent resources on the Golden Rectangle. Its pervasive presence in nature might demonstrate why it along with the golden spiral have fascinated artists for millennia. It fascinates me as an artist but also as a Mathematician. There are critics that say that it is overblown, but no one criticizes the beauty of the equation surrounding the golden spiral.

Google “golden rectangle” and look at the Wikipedia results for more information.
Also, see the following for more information about the golden rectangle or golden spiral – along with the Fibonacci Sequence (on my blog):

https://karenhalbert.blogspot.com/ 

https://karenhalbert.blogspot.com/2023/06/whirling-squares-spiral.html

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Root Rectangles inscribed in a Square (advanced)

A useful diagram for inscribing root rectangles within a square is:



The “x” in the equation is the gray vertical line from the intersection of the orange horizontal line and the quarter circle with radius 1 (1 is the hypotenuse for the smaller, similar triangle – similar to the full triangle with hypotenuse Root 2 and two legs of 1). Other Root horizontals: continue looking at the intersections of the quarter circle with the dotted lines (hypotenuses) for the additional Root horizontal lines.

The Root Rectangles 2,3, 4 and 5 as noted are of height 1/Root 2, 1/Root 3, 1/Root 4 and 1/Root 5 respectively. They all have width of 1. The proportion of the Root N rectangle is therefore 1 to 1/Root N, which is the same as Root N to 1. For example, if we multiply 1 and 1/Root 2 each by Root 2, then the resulting multiples have the same ratio:

Root 2 to 1.

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Detailed Construction of Douglas Fryer’s Square Armature with overlapped Dynamic Rectangles

The example of the painting in the video by Douglas Fryer may be studied with this armature as he demonstrated in the video.


  • Main Diagonals (dotted here).
  • Center Ver/cal and Center Horizontal
  • Quarter Circles (arcs) from each corner
  • Inner Square passing through intersections of arcs (and center verticals and horizontals)


Orange horizontal and Turquoise horizontal through intersections of the arcs and the main diagonals. (Root 2. See proof in Mathematics Appendix). These create overlapped Root 2 rectangles within the square.

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  • Clarification: Yellow and Green Verticals to create overlapped rectangles with aspect ratio around 1:1.16. These appear to be near golden rectangles.


  • Correction: Purple horizontals to create overlapped Root 3 rectangles with aspect ra(o 1:1.732.

    The subdivision into Dynamic Rectangles adds interest to a composition based on this armature. The artist may select one of the horizontal dynamic divisions as the horizon, with others as mountain lines or other landscape features lying on a horizon.

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Harmonic Armature Proportions and an Extension of Rule of Thirds

B x A, height B and width A Intercepts on the edges at half height and width



a1 is intersection of two lines.:

1) Y=B/(A/2)*x

2) Y = ((B/2)/A)*x + B/2

Set the y coordinates equal to each other and solve for x: B/(A/2) * x = ((B/2)/A)*x + B/2
2B/A * x = B/2A * x + B/2
(2B/A – B/2A)*x = B/2

(3/2)*(B/A)*x = B/2 3/A*x = 1 or x = A/3

Therefore, a1 is 1/3 of A, the distance from left edge to right edge.

a2 is intersection of two lines:

1) Y = ((B/2)/A)*x + B/2

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2) Y=-B/(A/2)*x+B

Set the y coordinates equal to each other and solve for x: ((B/2)/A)*x + B/2 = -B/(A/2) *x + B
(B/2A)*x +(2B/A)*x = B/2
((1/2A) + 2/A)*x = 1⁄2

(1/A) * (1/2 + 2)*x = 1⁄2
(1/A) * (5/2)*x = 1/2
(5/2)*x = A*1/2; multiplying both sides by A
5x = A or x = A/5; the x-coordinate of the point a2 is 1/5 of A. Therefore, a2 is 1/5 the distance from left edge to right edge.

Use the same process to find a3 and other intersections in the Harmonic Armature. It can be shown that all the intersections are at rational (fractional) divisions of the rectangle (panel).

The intersection of the main Baroque and Sinister Diagonals is in the center. Show this using the two equations indicated.

a3 is 1⁄4 of the width by inspection – but the interested student may show that it divides the canvas in fourths using the same method as for a1 and a2.

Where would the 1/6 mark be? Note there are other proofs.

A corollary to this is that a Root 2 Rectangle is divided into thirds by the verticals through the intersections of the main diagonals and their reciprocals. This is based on the division of a Root 2 rectangle into two vertical Root Rectangles meeting at the midpoint of the edges. See the section on the Comparison of Dynamic Symmetry (basic) Armature with Rule of Thirds Additional Examples.

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Miscellaneous Grids for Printing

Grids within a 3x4 box (popular camera size). Roughly in order of Aspect Ratio from Square (1:1) up to Root 4 (1:2) to Root 5 (1:2.34)

Square. Ratio 1

Double Root 4 Dynamic Symmetry


SQUARE “FRYER” ARMATURE




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Ratio 1.2xx





4 to 5 BASIC DS (RATIO 1.25) 11 TO 14 BASIC DS (RATIO 1.2727.. )


Root PHI Basic DS (Ratio 1.1272) Root PHI HARMONIC ARMATURE

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3 to 4 Ratio 1.333


BASIC DS Ratio 1.333 EXPANDED DS Ratio 1.333 With Horizontals

Harmonic Armature Ratio 1.333

Root 2. Ratio 1.414



2 to 3 Panel. Ratio 1.5



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Golden Ratio. 1.618 






____________________________________________________________

ROOT 3: Ratio 1.732



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ROOT 4. Ratio 2.0. 1 to 2 Proportion (eg; 8x16 or 10x20):



NEW:









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Root 5. Ratio 2.234



















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Select Bibliography

Hyperlinks not all included here (yet)

For another list, visit:

https://karenhalbert.blogspot.com/2023/05/reference-bibliography.html

Books and Artist Videos

  • Painters-Secret-Geometry-Study-Composition by Charles Bouleau. From an excellent review by T. Campbell: “This book certifies beyond reasonable doubt that the "intuitions" of the great European masters and well trained artists into the 20th century incorporated a healthy dose of geometrical technique to organize their image spaces harmoniously”. I recommend reading the whole review in this link to Amazon.

  • The Elements of Dynamic Symmetry by Jay Hambidge, the original guru of Dynamic Symmetry.

  • The Art of Composition. A simple Application of Dynamic Symmetry by Michel Jacobs

  • Mathematical Thought from Ancient to Modern Times by Morris Kline, 1972. This book was the textbook used for Liberal Arts’ majors at NYU.

• Creative Illustration by Andrew Loomis. Informal Subdivision, pgs. 36-39.

  • Geometry of Design by Kimberly Elam. Beautiful little book. The transparent over leaves are a real plus. A good one to read if this is the only one you read. It does promote the golden mean as a design principle.

  • Composition by Arthur Wesley Dow

  • Composition of Outdoor Painting by Edgar Alwyn Payne

  • Intuitive Composition by Albert Handell

  • Video on Rocks and Water by Handell: Rocks and Water,

  • Video by Douglas Fryer (short Version):

    https://www.facebook.com/PaintTubeTV/videos/1191144498112822(this shortened

    free version does not have the Composi2on sec2on)

  • Full Length Video by Douglas Fryer from Streamline: Painting with Intution:

    https://painUube.tv/products/douglas-fryer-paintng-with-intuition

  • Youtube Video by Michele Byrne: Using Dynamic Symmetry in Art

• https://www.goldennumber.net/art-composition-design/. Meisner. Major website on the golden number.

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The Science of Beauty Article:

Article recently published on The Science of Beauty, by Bob Bahr, which includes artwork and Dynamic Symmetry by Halbert, Fryer and Byrnes. 

As of 11/18/2024 flip book view: https://online.flippingbook.com/view/354061990/ (but to be removed)

Golden Mean Videos:

Websites with extensive material on Armatures:

• Of interest might be the 8 minute video by Tavis Leaf Glover on constructing the Dynamic Symmetry Armature in Photoshop: 

https://www.iso1200.com/2017/11/dynamic-symmetry-how-to-quickly-build.html

(view if you have any experience with photoshop but it might be helpful in any case).


https://youtu.be/lluL6tuyif8. Simple but very clear for beginners. 

https://www.youtube.com/watch?v=cZ_SlaH3fA0 with the true meaning behind the math.

• ALSO Youtube video by Tavis: https://www.youtube.com/watch?v=PaUsB57UF1Y

Participating Artist Websites:

Evelyne Boren: https://evelyneboren.com/
Carole Belliveau: https://www.carolebelliveau.com/
Michele Byrne: https://www.michelebyrne.com/
Elizabeth Ming Cooper: https://www.elizabethmingcooper.com/ 

Douglas Fryer: https://douglasfryer.blogspot.com/ and
Bill Gallen: https://www.billgallen.com/
Karen Halbert: https://karenhalbert.com/
Albert Handell: https://www.alberthandellstudio.com/
Cynthia Inson: https://www.cynthiainsonart.com/
Natasha Isenhour: https://www.natashaisenhour.com/
Thomas Kegler: https://thomaskegler.com/

81

Mathematics Appendix: Self-Similar Rectangle

Theorem

Assumption: the slope of a line perpendicular to a given line is the negative reciprocal.

Theorem: the reciprocals to the main diagonals mark off vertical lines that form new (vertical) rectangles proportional to the original rectangle. Proof: Mark off a rectangle as indicated using the (x,y) coordinate system.



Now mark off a vertical line from the intersection of the reciprocal at the bottom edge at (w1,0).



Substitute (w1,0) into the reciprocal line equation to obtain



82

Addendum:
Constructing Golden Mean Calipers

Links for making your own Golden Mean Calipers: Video: How to Make Your Own Golden Mean Calipers. https://www.youtube.com/watch?v=yxXI5a6B79s&t=44s

Website: https://www.goldennumber.net/do-it-yourself/

83

Construction of the Golden Rectangle

(see 2025 Diagrams)



Figure 1 Unit Square



Figure 2 Unit Square with line from midpoint to upper right corner

Find the midpoint of the bottom of a 1x1 square as indicated.
Draw the line from this midpoint to the upper right corner of the square.

84




Figure 3 Unit Square with Arc from upper right to bottom edge

Construct the partial circle with radius the length of this line to the bottom edge (extension) with a compass (or a taut string with pencil).



Figure 4 Golden Rectangle with Unit Square and Vertical

Complete the rectangle formed by the square edge of height 1 and the lower edge to the end of the arc as indicated.

This is the Golden Rectangle.
See the section on Golden Rectangle Measurements. This rectangle will be shown to have

the proportion 1 by 1.618..., the Golden Mean.

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Measurement of the Golden Rectangle





Figure 5 Golden Rectangle with Unit Square and "radius" measurements

What is the radius, r?

From the Pythagorean Theorem:
r *r = .5 * .5 plus 1*1 = .25 + 1 = 1.25
r = square root of (1.25) = 1.118... (using a calculator).

Add 0.5 to 1.118 to get 1.618, the width of the full rectangle.

Hence the proportion of the full rectangle is 1 x 1.618.... 1.618 is the golden mean (approximately)!!!

Notice the new vertical rectangle to the right of the unit square.
What is its proportion? Let’s lay it “flat” so that its height is r – .5 = 1.118 - .5 = ,618!!!. Then its proportion is .618 by 1!

The new vertical rectangle has proportion 0.618 x 1 (rotated).

What is 1 divided by .618 (take out the calculator): It is 1.618!! 


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