Definitions

Orthogon

Orthogon

Orthogons as Design Templates


Or"tho*gon (?), n. [Ortho- + Gr. angle: cf. F. orthogone, a.] (Geom.) A rectangular figure OR'THOGON, n. [Gr. right, and angle.] A rectangular figure. (Webster's Dictionary)

Artists, architects and calligraphers for centuries have used a set of templates to guide the placement of elements in their designs. In 1956, glassware designer, Wolfgang Von Wersin, referred to these templates as "orthogons". His book was titled, The Book of Rectangles, Spatial Law and Gestures of The Orthogons Described (Das Buch vom Rechteck Gesetz und Gestik des Raumlichen die Othogone-scheibe--Die Orthogone-scheibe), Otto Maier Verlag Publishers, Ravensburg, West Germany.

The most famous orthogon, the Auron (Golden Section, Golden Ratio, Golden Mean, Golden Rectangle) has been used at least since the time of the ancient Greeks to design everything from the Parthenon to calligraphy to ceramic vases.

All twelve orthogons function as an archetype in that they are both "form" and "content". "Form" because they play a role in composition and "content" by the various meanings that can be assigned to "order", "relationship", "harmony", etc. Use of the orthogons conforms to the definition of art offered by Clive Bell: "significant form."

Some of the orthogons are related to musical harmonies.

According to Wolfgang Von Wersin, “The Orthogons are without exception root figures and are all irrational numbers. The calculation of the measure relations within the Orthogons are based, without exception, on the Pythagorean doctrine.” (pg 80.)

Constructing an Orthogon

Orthogons always begin with a square, any square. Individual orthogons are constructed and the measurements are then used to guide a design.

Diagon

The entire series of 12 orthogons can be described by choosing a square, which is expanded through a series of extensions (scribing arcs outward) until another square is formed on top—an exact duplication of the original square. The good news for artists is that orthogons can be created without using a ruler or precise numbers as a guide for designs. The Egyptians tied knots on a rope to re-plot the land annually flooded by the Nile. Similarly, Orthogons can be created using:

• Lengths of a string (similar to Greek methods)

• A compass

• Marks drawn on the straightedge of paper or stick

The orthogon understructure need not be apparent in the work but the arcs, diagonals, squares, etc. are used to direct a concept or solidify a design.

The orthogons relate to root numbers and their measurements are irrational numbers. Ratios for all twelve orthogons:

Instructions for constructing four of the less-complex orthogons can be found at

Diagrams for all 12:

Application in Design

“All art stands in the vital tension between the part and the whole. And without an anticipating vision of the whole, no true and reliable picture or knowledge of the part is possible.” Hagen Haltern

Orthogons serve several vital purposes in that they provide:

• Ideas for where to begin a design and how to proceed

• A format for elements to relate to each other and to the work as a whole

• Potential layers of meaning—elements relate to each other philosophically and physically

Typically only a few simple measurements for use within a work are necessary (small, medium and large) to solve the planning mysteries.

Orthogons may have been applied by such artists and architects as: Brunelleschi, Mondrian, Georges Braque, Degas, Mark Rothko and they certainly were used by Giorgio Morandi, whose colleagues chuckled at his constant “measuring”.

At first glance, Morandi’s collections of bottles and vases appear to be just that, simple collections. A closer look reveals an amazing array of possible concepts orchestrated within repeating spaces every direction, even down to the size and placement of his name.

Walter Valentini, a contemporary Italian artist, defines and highlights Orthogons in molded paper, color and gold leaf.

Historical Use

"Cathedrals, like other sacred structures, were designed not by whim, taste or fashion, but according to the timeless principles of sacred architecture and the archetypal patterns which emerge from number and geometry.” Michael Schneider

The character in the painting “Allegory of Geometry”, (1649 by Laurent de la Hire) holds a compass and a paper with a configuration that gives reference to the approach typically used when forming Orthogons. The character drawn in the painting is the Auron or Golden Section.

The Parthenon and Pantheon are believed to have been constructed using the Golden Section’s series of relationships to make a unified structure related to religious beliefs. Many artists in this and previous eras have also incorporated design structures such as the Orthogons. According to Jay Hambidge’s excellent discourse on Dynamic Symmetry:

“The principles of design to be found in the architecture of man and of plants have been given the name ‘Dynamic Symmetry’… The Greeks obtained knowledge of dynamic symmetry from the Egyptians some time during the 6th century B.C…In Greece, as in Indian and in Egypt, the scheme was connected with altar ritual…The Greeks, however, soon far outstripped their Egyptian masters and within a few years…apparently made the astounding discovery that this symmetry was the symmetry of growth in man.”

According to recent brain research, humans are most interested in designs that have a high degree of both order and variety. Brain waves are stimulated by the combination of both order and variety in that they flatten with too much order (boring) and they also flatten with too much variety (confusion). Use of a design template accomplishes both: a high degree of variety through the images and techniques that are applied to the high order of a template.

Sacred Geometry, Sacred Art

Natural forms often follow a general pattern related to geometry (phi). This consistent appearance of pattern has been viewed as awe-inspiring and even sacred. The consideration is that if man discovers and applies the system that God uses (sacred geometry), then what is created has eternal connections.

With respects to sacred art, Keith Critchlow states:

“Just as the Word of God is sent down to a specific people in a language and form that is intelligible and compelling to them, so, too, reciprocally, must the practitioner of sacred art utilize a particular 'language' and form to consecrate his or her work in order that it might ascend to rejoin its heavenly archetype and 'transport' other believers in the community there as well.” Certainly designers drafting everything from buildings, cars, calligraphy, quilts and logos are aware of the subliminal interest generated by a particularly unique orthogon, the Auron.

Auron (Golden Section)

The Golden Section was identified by previous generations as the system of order for this world. Creators, from the Parthenon to Renaissance cathedrals, understood that the psychological power of this specific order serves as a subconscious language in line, dimension and space. The Golden Section also has connections with the Fibonacci Sequence (1, 2, 3, 5, 8, 13, etc. each number adds to the previous to arrive at the next), a pattern found in abundance in the natural world.

One of the reasons the Auron holds such fascination is that within every Auron is another Auron--infinity either direction.

Unless the measurements of the Auron are skillfully applied (within a square or a short form), they tend to be long and a bit unwieldy for design purposes. The additional orthogons allow for a greater range of design choices.

Application of the orthogons may have been confined to guild use although connections have been identified with sacred geometry, artwork of cultures around the world and musical harmonies.

Auron

Orthogon Instructions

Orthogons, although they may seem tedious at first, save time in the long run by guiding the placement of various elements such as line, shape, proportion, even color.

Once an orthogon is formed, several relationships automatically are available to use. Diagrams in countless books typically refer only to the basic diagram of the orthogon, which is of little use in actual application.

The original basic design of an orthogon has only two measurements—the square and the portion beyond the square. The two divisions can be applied in a variety of directions but are usually too static for most compositions.

Consider the variety of space divisions (including a small measurement) in the color fields by Mark Rothko.

Applying an Orthogon to a design 1. Decide and mark the measurement of what will be the shortest edge of the final work (can be done with a string, paper or board).

2. Create a square using the edge measurement (yellow line on the diagram below).

3. Create an Orthogon from the square (the orthogon is the final rectangle created).

4. Divide the square into quarters (light green line).

5. Starting in the top left quarter, measure from the bottom corner of the quarter diagonally up to the top edge of the same quarter.

6. Lay that measurement down on the top edge of the square and mark (dark blue line).

7. Start in the same bottom corner as #5 and measure up to the new mark of #6.

8. ay that measurement down on the top edge of the square and mark (dark green line).

9. Start in the same corner as #5 and measure to the mark of #8.

10. Lay that measurement down on the top edge of the square and mark (orange line). If done correctly, this measurement will be in alignment with the square.

11. Set the edge of a paper (or wooden stick) along the top edge of the orthogon and mark all the new distances (including the dark purple section extending beyond the square).

Order and Variety

Now comes the final point of departure from the high degree of “order” the orthogon affords, where “variety” has full reign. Every maple leaf looks like--a maple leaf, but no two in the entire world are exactly alike.

Lay down the various elements of the work, aligning them in relation to marks posted on the edge (as designated by the chosen orthogon). All of these edge marks will relate to each of the other marks. Diagonals or arcs can “point” from one edge mark to another. The marks can extend beyond the edge of the work out into the main body and remain visible in the artwork. Portions of one element may relate to an ending point or line of another shape, including your signature!

Measurements relate to each other and to the work as a whole; side caption reads: "Distance halfway on the square"

The short side of the rectangle can be used to identify the square

Excellent examples of this procedure can be determined by analyzing the artwork of Mark Rothko and Giorgio Morandi. The image below was created using a Quadriagon. The square can be identified by measuring the short section of the work. Determine additional measurements and notice how they are repeated.

Joan's Offering

Valrie Jensen

10"X13" Etching/Monoprint

Quadriagon

When applying an orthogon to unusual designs, such as a slender knife, the small/medium/large measurements can be used--any direction.

More detailed explanations and illustrations can be accessed through the website, timelessbydesign.org:

Conclusion

“Geometric organization in and of itself does not yield the dynamic concept or inspiration. What it does offer to the creative idea… is a system of bringing the elements together into a cohesive whole.” Kimberly Elam

Artists and architects have noted the “wonderful sound” of ancient Greek architecture. Perhaps the harmonious relationships of these buildings, as well as the dimensions of subsequent Roman architecture, actually create an audible resonance.

Filippo Brunelleschi, with the aid of his assistant, Donatello, was likely the first Italian Renaissance artist to study the physical dimensions of Roman Architecture. Brunelleschi's elegant designs align with the dimensions of the orthogons and include: Foundling Hospital in Florence, Italy (Ospedale degli Innocenti), the Dome (Duomo} for the Cathedral of Florence, and two Florentine churches, Basilica di San Lorenzo di Firenze, and Santo Spirito di Firenze. The graceful arches and harmonious features have a unique quality of resonance as well, which may support the enduring appeal of structures designed by Filippo Brunelleschi.

Wolfgang Von Wersin's book includes an extraordinary copy of text from 1558 printed in Nurenburg Germany, with diagrams of seven orthogons and an invitation from the passage to pay careful attention as the “ancient” architects believed “nothing excels these proportions” as “a thing of the purest abstraction.” (pg 36.)

"More art, faster!" Wulf Barsch

All 12 Orthogon Names

Quadrat

Hemidiagon

Trion

Quadriagon

Biauron

Penton

Diagon

Bipenton

Hemiolion

Auron

Sixton

Doppelquadrat

References

Bulent Atalay, Math and the Mona Lisa: The Art and Science of Leonardo da Vinci; 2006, Smithsonian Institution, HarperCollins Publisher, NY, NY.

• R. A. Schwaller de Lubicz, The Temple in Man; 1977, Inner Traditions International, Rochester, Vermont.

• Gyorgy Doczi, The Power of Limits, Proportional Harmonies in Nature, Art and Architecture; 1981, Shambala Publications, Inc., Boston, Massachusetts.

Albrecht Durer, Of the Just Shaping of Letters, From the Applied Geometry of Albrecht Durer Book; Dover Publications, NY, NY.

• Kimberly Elam, Geometry of Design: Studies in Proportion and Composition; 2001, Princeton Architectural Press, NY, NY. ISBN 9781568982496

• Hagen G. Haltern, Art Integration, The Spiritual Foundation and Anagogical Level of Meaning of the Celestial Style, 1989, H and H Book, Orem, Utah.

Jay Hambidge, The Elements of Dynamic Symmetry; 1967, Dover Publications, NY, NY.

• David Harris, Calligraphy, Modern Masters—Art, Inspiration and Technique; 1991, Crescent Publishers, NY, NY.

• Valrie Jensen, Timeless By Design; 2007, Willow Creek Art Center, Sacramento, CA.

Mircea Eliade, Cosmos and History, The Myth of the Eternal Return; 1959, Harper & Brothers Publishers, NY, NY.

• Anthony Lawlor, AIA, The Temple in the House; 1994, G.P. Putnam’s Sons Publisher, NY, NY.

Hugh Nibley, Temple and Cosmos; 1992, Deseret Book Company, Salt Lake City, Utah.

• Neri Pozza, Morandi: Dessins, Drawings; 1976, Ideae, Milan, Italy.

• Michael S. Schneider, A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science; 1994, Harper Paperbacks. ISBN 0-06-092671-6

• Robin Langley Sommer, Frank Lloyd Wright; 1998, World Publications Group, Inc., North Dighton, MA.

• Wolfgang Von Wersin; Die Orthogon-scheibe: Das Buch vom Rechteck Gesetz ungestik des Raumlichen die Othogon-scheibe; 1956, Otto Maier Verlag Publishers, Ravensburg, Germany.

Further Reading

• Samuel Colman, Nature’s Harmonic Unity, a Treatise on its Relation to Proportional Form; 1971, Benjamin Blom, Inc., Publishers, NY, NY.

Keith Critchlow, Order In Space: A Design Source Book; 1970, Viking, NY, NY.

Morris Kline, Mathematics for the Nonmathematician; 1967, Dover Publications, Toronto, Canada.

Robert Lawlor, Sacred Geometry: Philosophy and practice (Art and Imagination); 1989 Thames & Hudson. ISBN 0-500-81030-3

Karl Menninger, Number Words and Number Symbols, A Cultural History of Numbers; 1969, Dover Publications, English translation copyright, Toronto, Canada.

• Stanley Morrison, Pacioli’s Classic Roman Alphabet; 1994, Dover Publications, Inc., Toronto, Canada.

Mark Rothko, Mark Rothko; 1996, Tate Publishing. ISBN 1854372122

George Santayana, The Sense of Beauty, Being the outline of Aesthetic Theory; 1955, Dover Publications, Inc., NY, NY.

06:46, 15 September 2008 (UTC)

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