Check out examples of works that the students have created over the past Sacred Geometry class sessions. I've kept their works as anonymous to respect their privacy. Enjoy!

**AA1o**

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PCA Sacred Geometry

## Monday, June 14, 2010

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Students Interpretations

## Wednesday, May 27, 2009

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Class 4 and 5

## Wednesday, April 15, 2009

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Class 2 and 3

## Saturday, March 21, 2009

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Sacred Geometry

**Welcome students and visitors of Sacred Geometry!**

I have setup this page to promote the class before it begins on Thursday March 26th, but mostly as a during-class dialog between the students, visitors and I. Therefore, the tone here is addressed to the students, as if the first class has already happenned. You will still be able to get a sense of what the class will cover though, at least during its introduction. This blog is made so that we can communicate to each other and share ideas about the class, comments, questions and projects. I am just learning about this blogging thing so I'm going to be experimenting a bit...

## Alberto's Blogs

## Blog Archive

## Alberto's Geometries

A technical and conceptual class by Alberto J. Almarza Pittsburgh Center for the Arts 8 Thursdays 6:30 to 9:30

This is a detail from a Mandala design by a dear continuing Sacred Geometry student Christianna Kreiss. Since taking her first class, Christianna continues to create lots of beautiful Mandalas, Rose windows and other designs.

Check out examples of works that the students have created over the past Sacred Geometry class sessions. I've kept their works as anonymous to respect their privacy. Enjoy!

**AA1o**

Check out examples of works that the students have created over the past Sacred Geometry class sessions. I've kept their works as anonymous to respect their privacy. Enjoy!

These two classes were devoted to a specific topic each. Class 4 was mostly dedicated to the topic of Geometry in Nature (as a continuation of class 3), while class 5 was focused on Geometry in the Man-made World. In general, we learned that Sacred Geometry deals with both, and that in fact the principles are correlated, showing precisely the connection between the work of humans and their inspiration from nature, as being part of one unifying principle: that of "Heaven on Earth."

In the "Geometry in Nature" class, we made many connections between our drawing surface and the world around us. After looking at the seemingly familiar world of plants, we saw examples of geometry found in nature that were beyond what our own senses can perceive, looking at images both microscopic and telescopic. We saw galaxies, clusters, planetary orbits and rings, down to river banks, craters, clouds and huracanes, down to animal skeletons, seashells and even further down and inside, to organs, arteries, cells, atom collisions and Quanta! (just kidding, we didn't get that tiny, but we did see atom collisions!). All of these forms shared many of the familiar geometries we have been studying, and the concepts and symbols associated with these patterns seemed to make sense across scales. For example, we talked about the spiral being sometimes associated with the notion of "growth" and experienced precisely this sense when looking at examples in the natural world, at many different scales. Galaxies, plants and seashells are spiral because of their growth, no matter their size. We also experienced some curious connections, such as learning that the eye of a bee is made of a hexagonal matrix, very similar to the pattern of a beehive.

In the "Geometry in Nature" class, we made many connections between our drawing surface and the world around us. After looking at the seemingly familiar world of plants, we saw examples of geometry found in nature that were beyond what our own senses can perceive, looking at images both microscopic and telescopic. We saw galaxies, clusters, planetary orbits and rings, down to river banks, craters, clouds and huracanes, down to animal skeletons, seashells and even further down and inside, to organs, arteries, cells, atom collisions and Quanta! (just kidding, we didn't get that tiny, but we did see atom collisions!). All of these forms shared many of the familiar geometries we have been studying, and the concepts and symbols associated with these patterns seemed to make sense across scales. For example, we talked about the spiral being sometimes associated with the notion of "growth" and experienced precisely this sense when looking at examples in the natural world, at many different scales. Galaxies, plants and seashells are spiral because of their growth, no matter their size. We also experienced some curious connections, such as learning that the eye of a bee is made of a hexagonal matrix, very similar to the pattern of a beehive.

These connections are inumerable and they give us a sense of wonder. Is it all random, or is it all connected?

This question led us to deeper terrains and we explored, for example, the idea of a connection existing between animate and innanimate matter. We saw that, at least according to the forms of sacred geometry, living and dead matter are very much connected by the notion of *pattern*.

We briefly looked at one the first books proposing this idea, and even though now many of the postulates in this book are obsolete, we found some very valid points. The book is called "Of Growth and Form" by D'Arcy Thompson. This was one of the first books of this kind, and it reformed the way biology is conducted and understood. In a nutshell, Thompson argued that the patterns and forms of living beings were the direct material imprint of physical and chemical processes. He argued that organic form was a result of the same innanimate processes one could study in the lab. In class, we conducted one of the experiments described in the book, that of simply releasing a drop of ink in a container of water. I didn't want to bias the class, so instead of saying much, after I released the drop I asked: "What do you see?" One of the students spontaneously replied: "Jelly fish!" It was fascinating for the class to see that in his book, D'Arcy Thompson had the drawings of the ink drop experiment inmediately along side jellyfish. His point was made.

But is there a deeper connection between animate and inanimate matter, other than pattern? Ok, so, the sun is not alive and it is round and an eye is live tissue and it is round too, so they are connected. Big deal. Blood vessels and river beds look exactly alike, one is giant one is tiny, one is live the other is dead, they are connected by pattern... big deal.

We learned about another interesting idea to promote the notion of wholeness, or interconnection between all things (in this case live and innert matter). This is a reocurring theme in our study of Sacred Geometry, because remember, a big part of our definition involves the idea of a **connecting principle**, one that relates things across scales, disciplines and philosophies. It is a theory proposed by James Lovelock and Lynn Margulis called GAIA theory.

Unlike popular belief, Gaia theory is not a hippie or new-age idea relating to the Goddess Mother Earth being alive. This is, rather, just a convenient metaphor to describe the theory in a few words! Gaia theory is an actual scientific development, and one that is crucial to our survival in this planet (the planet itself is fine, it is US who are in danger!)

Gaia theory proposes that, instead of thinking of the Earth as a dead planet, made of rock, ocean and atmosphere, merely inhabited by life on its surface, that planet Earth is a self-regulating system, and all of its elements and components actively participate in the continuation of the entire system. The planet is thus considered to be closer to a living organism than it is to a rock, in that it can self-organize and regulate itself. This is not to say that Earth is intelligent (at least not in the antropomorphic sense) but it is to say that every rock, bug, cloud, tree and person, ALL play a huge roll in the development and continuation of life on this planet. Water is not alive(I'm not even sure about this :)), and yet without it life is not possible. I can't go too in depth here, since that is the point of taking the class, but we did cover one good example used by Lovelock and Margulis to support and explain their theory in easy terms. It is the cycle of CO2 as described by GAIA theory. I will just direct you to an awesome book by Fritjoff Capra called "The web of Life". In this book, he describes the cycle of CO2 as a good example of the collaboration between animate and innanimate matter to sustain life on Earth. He also describes Gaia theory, and covers very digestable explanations of systems theory, cybernetics, quantum physics, self organization, genetics, mysticism and many other things which are key to a new understanding of the universe. WOW! Just get the book...

In our "Geometry in the Man-made World" class, we talked about humans deriving inspiration from nature to create sacred sites and buildings, monuments, designs, icons, symbols and even everyday products! We discussed how the same principles found in nature such as the Golden Ratio, the circle, the polygons, proportions, etc, were used as the basic structures for creating objects. In this sense, we entertained the ancient idea that using these principles can create a bridge between the material, physical world and the realm of the Heavens.

We looked at architecture, and how sacred geometry plays a central role in creating every part of a building, from its floor plan to its elevation, through all its details and ornaments.

We looked at art, tiling and tesselation, mandalas and mechanisms, all as examples of the use of mathematics, geometry and pattern, for the creation of beautiful, spiritual, or efficient objects.

So we discussed "Biomimicry", which literally means "imitating life", as a widely used principle in art, engineering, architecture and science. I want to share here a video we watched in class about artist/inventor Theo Jansen and his "animals". This is a great example of biomimicry, and all the more related to our class since he mentions the use of "eleven holy numbers", which are totally mysterious to me. I've heard someone say, very matter of factly: "Its most likely the Fibonacci sequence" but this makes absolutely no sense, since by the time you get to the eleventh digit, 89, the difference in measure would be so radical to the first digit, 1, that you could not make this mechanism work. Anyway...

I also wanted to share this beautiful drawing by Architect Louis Sullivan. I wanted to show, on one hand, his enourmous understanding of how to describe organic forms with drawing tools, but mostly because it came up in our class that asymmetry is as beautiful as any form of symmetry, which is something that became an important topic in our fractal and Li discussions.

So, it seems as though we have gotten quite far along and we are now past the introduction sessions, going more in depth. In class 2 we covered a lot of important basics such as some definitions of beauty and symmetry and a basic vocabulary of number symbolism.

To define beauty we discussed the thoughts of many philosophers who have dealt with the idea, trying to identify a thread in the notion of beauty throughout history. Plato, Aristotle, Epicurus, Aquinas, st. Augustine, De L'Orme, Coleridge, Birkhoff, Huntley, Tufu, Buckmisnter-Fuller and many others were quoted. We found several important notions and keywords: order, harmony, symmetry, balance, repetition, exactness, perfection, simplicity, complexity, wholeness, clarity. We discovered that beauty is attributed both to the object AND to the "eye of the beholder" or the subject. In other words, beauty can be both objective and subjective. We also understood that, while beauty can be immediate and in the surface, its depth can increase considerably with experience, study and education. For example, we looked at a text from H. E. Huntley's Divine Proportion: "The surface beauty of the rainbow is appreciated by all men, it is given. But its buried beauty, uncovered by the industrious research of the physicist, is understood only by the scientifically literate. It is aquired: education is essential". This is an important notion in our class. At some point, I said: "flowers are beautiful, yes, we all agree. But once we have completed these studies, a flower may just take your breath away". On the first class, I also commited to giving all of you "sacred geometry goggles", right? I think by now you are all begginning to understand what I meant.

We also covered the four basic categories of symmetry: Translation, reflection, rotation, and glide-reflection. I briefly mentioned self-similarity as being another important form of symmetry which we will cover in depth once we get to our fractal class.

We had some "dimensional travel" as we reviewed the basic "essences", symbols and associations related to numbers from 0 to 5. Also, as our knowledge gets deeper, our constructions have gotten a bit more complex too. By now, you should all know how to draw, with precision and at any given scale, the following elements of Sacred Geometry:

**Vesica Pisces**

**The flower of Life **

**The Torus**

**Circle and Phyllotaxis Matrices**

**Regular polygons**

**The Golden Ratio: segment, rectangle, spiral.**

With these constructions, you now have a large vocabulary of Sacred forms to contemplate, draw, study and have fun with. Needless to say, what you can do creatively with these forms is unending. I encourage you to experiment and play with the constructions, wether it is making mandalas and abstract images, analizing and justifying nature objects, designing architectural features and objects or simply getting practice with your compass. Change a thing here and there, see what happens when you revert the order of steps, connect different forms, explore and experiment. You will be amazed to see how connected to each other these forms can be, and once you start making connections you will find that really, all these forms are VERY closely related, visually and metaphorically. Remember, we are always trying to see the multilayered nature of these forms, both in their immediate shape and in their deeper meanings and connotations. Here is an example of something I did, connecting the pentagon, the golden isosceles and the golden spiral. We will get into more advanced constructions and connections as we go.

I also wanted to briefly recall here a bit about what we discussed regarding Phyllotaxis. The whole of class 3 was devoted to the topic of**geometry in nature**. Hence, the form and features of the plant kingdom were greatly discussed. Phyllotaxis literally means "plant growth". We learned that the entire range of forms in the plant kingdom can be studied through the lense of phyllotaxis. We covered the four basic types of phyllotaxis: distichous, whorled, spiral and multijugate. We discovered that most plants feature spiral and whorled phyllotaxis. More importantly for our class, we learned that, in one particular study, out of the vast amount of plants that displayed spiral and whorled phyllotaxis it was found that 92% of them showed fibonacci phyllotaxis! This means that the Fibonacci sequence and our golden number PHI are very much a big part of the plant world.

Briefly, the Fibonacci sequence involves numbers that are in a series where any number is the sum of the two preceding numbers. Therefore we have:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

As we approach the fifteenth digit, dividing a number in the sequence by the previous number gives us a result which is closer and closer to PHI= 1.61803...

As you learned in our Golden Ratio lesson, this is the "magic" number that is so crucial to the entire development and practice of Sacred Geometry. This is the reason why phyllotaxis (which is also in connection with the torus) is an important development to keep in mind. So next time you see a pinecone or a sunflower, go ahead and count the "parastichies" or spirals, and you will find that they are two consecutive numbers in the Fibonacci sequence, almost ALWAYS.

Now, in this pinecone you see 13 parastichies going clockwise and 8 parastichies going counter-clockwise, a beautiful example of fibonacci phyllotaxis! But remember, these are just the "visually obvious" examples of phyllotaxis, for it can be seen even in the general leaf arrangement of almost any plant. If you are interested in this, I highly recomend you visit this link, the official phyllotaxis website, full of details, information and a beautiful gallery of images.

Well students and visitors of Sacred Geometry, I will continue posting ideas; I hope you find this useful in some way.

To define beauty we discussed the thoughts of many philosophers who have dealt with the idea, trying to identify a thread in the notion of beauty throughout history. Plato, Aristotle, Epicurus, Aquinas, st. Augustine, De L'Orme, Coleridge, Birkhoff, Huntley, Tufu, Buckmisnter-Fuller and many others were quoted. We found several important notions and keywords: order, harmony, symmetry, balance, repetition, exactness, perfection, simplicity, complexity, wholeness, clarity. We discovered that beauty is attributed both to the object AND to the "eye of the beholder" or the subject. In other words, beauty can be both objective and subjective. We also understood that, while beauty can be immediate and in the surface, its depth can increase considerably with experience, study and education. For example, we looked at a text from H. E. Huntley's Divine Proportion: "The surface beauty of the rainbow is appreciated by all men, it is given. But its buried beauty, uncovered by the industrious research of the physicist, is understood only by the scientifically literate. It is aquired: education is essential". This is an important notion in our class. At some point, I said: "flowers are beautiful, yes, we all agree. But once we have completed these studies, a flower may just take your breath away". On the first class, I also commited to giving all of you "sacred geometry goggles", right? I think by now you are all begginning to understand what I meant.

We also covered the four basic categories of symmetry: Translation, reflection, rotation, and glide-reflection. I briefly mentioned self-similarity as being another important form of symmetry which we will cover in depth once we get to our fractal class.

We had some "dimensional travel" as we reviewed the basic "essences", symbols and associations related to numbers from 0 to 5. Also, as our knowledge gets deeper, our constructions have gotten a bit more complex too. By now, you should all know how to draw, with precision and at any given scale, the following elements of Sacred Geometry:

With these constructions, you now have a large vocabulary of Sacred forms to contemplate, draw, study and have fun with. Needless to say, what you can do creatively with these forms is unending. I encourage you to experiment and play with the constructions, wether it is making mandalas and abstract images, analizing and justifying nature objects, designing architectural features and objects or simply getting practice with your compass. Change a thing here and there, see what happens when you revert the order of steps, connect different forms, explore and experiment. You will be amazed to see how connected to each other these forms can be, and once you start making connections you will find that really, all these forms are VERY closely related, visually and metaphorically. Remember, we are always trying to see the multilayered nature of these forms, both in their immediate shape and in their deeper meanings and connotations. Here is an example of something I did, connecting the pentagon, the golden isosceles and the golden spiral. We will get into more advanced constructions and connections as we go.

I also wanted to briefly recall here a bit about what we discussed regarding Phyllotaxis. The whole of class 3 was devoted to the topic of

Briefly, the Fibonacci sequence involves numbers that are in a series where any number is the sum of the two preceding numbers. Therefore we have:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

As we approach the fifteenth digit, dividing a number in the sequence by the previous number gives us a result which is closer and closer to PHI= 1.61803...

As you learned in our Golden Ratio lesson, this is the "magic" number that is so crucial to the entire development and practice of Sacred Geometry. This is the reason why phyllotaxis (which is also in connection with the torus) is an important development to keep in mind. So next time you see a pinecone or a sunflower, go ahead and count the "parastichies" or spirals, and you will find that they are two consecutive numbers in the Fibonacci sequence, almost ALWAYS.

Now, in this pinecone you see 13 parastichies going clockwise and 8 parastichies going counter-clockwise, a beautiful example of fibonacci phyllotaxis! But remember, these are just the "visually obvious" examples of phyllotaxis, for it can be seen even in the general leaf arrangement of almost any plant. If you are interested in this, I highly recomend you visit this link, the official phyllotaxis website, full of details, information and a beautiful gallery of images.

Well students and visitors of Sacred Geometry, I will continue posting ideas; I hope you find this useful in some way.

I have setup this page to promote the class before it begins on Thursday March 26th, but mostly as a during-class dialog between the students, visitors and I. Therefore, the tone here is addressed to the students, as if the first class has already happenned. You will still be able to get a sense of what the class will cover though, at least during its introduction. This blog is made so that we can communicate to each other and share ideas about the class, comments, questions and projects. I am just learning about this blogging thing so I'm going to be experimenting a bit...

I am imagining that you have been to the first class, and so by now you should have a basic idea of the overall definition of Sacred Geometry we are working with. We are talking about a field of study which is very wide-encompassing, covering areas of art, the sciences and humanities. It is also a field which explores forms, patterns and symbols that are very timeless, and reocuring throughout the man-made and natural world across many different geographic locations.

We talked about Sacred Geometry being multi layered in that it has a**technical** layer, a **practical** layer, and a **metaphorical** layer.

The**technical layer** involves the geometric, mathematical, physical, and scientific principles and laws that surround the many forms involved with Sacred Geometry. It also involves their construction, analysis, calculations, etc.

We talked about Sacred Geometry being multi layered in that it has a

The

The **practical layer** is related to the application of some these principles, whether it is in the man-made world: architecture, art, music; or whether it is as seen in the natural world in the form of biological structures, geographic formations, the atomic level, the astronomical universe, the human mind, etc.

The**metaphorical layer** involves the meanings, symbols, relations and concepts associated with the specific forms of Sacred Geometry. This is the layer that engages the field beyond the physical studies, onto the conceptual aspects of art, philosophy, religion, spirituality, mysticism, and many other transcendental practices.

The

Even though we will explore and discuss all of these topics, our class will place a lot of focus on the technical layer, since we are working on learning how to construct many of these geometric forms using only a compass and a straightedge, mostly for artistic purposes. We are looking at a variety of practical constructions and methods used since antiquity for generating these forms and graphical elements. I have been inspired and informed by the related work of artists and thinkers such as Leonardo Da Vinci and Albretch Durer, and more recently by M.C. Escher, Gaudi, Joseph Beuys, and many others.

In this first class, the constructions covered are the very basic building blocks first: bisecting a line, bisecting an angle, proyecting a distance, dividing the circle, etc. We are also covering some basic, more exciting and ornamental constructions, such as Vesica Piscis, the Flower of Life and the Torus. Along the line, students will have learned more complex constructions, such as regular polygons, Phyllotaxic matrices, tessellation, the golden rectangle and spiral, and some fractal designs.

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I begun studying the concepts of Sacred Geometry over ten years ago, when I would get together with my Aunt Emi and we would spend entire afternoons trying to figure out M. C. Escher's drawings. We called it "The Escher Workshop" and we would meet every Friday with our compasses and straightedges to draw tessellations, mandalas,polyhedra and impossible figures. She re-taught me the basic geometric constructions from school days, and with those tools as a starter, I begun delving deeper and deeper. Even though I have very little understanding of the mathematics behind geometry, I have made a point to develop an entire vocabulary of constructions, and specifically ones that would help artists to create artful geometric forms at any scale. I use mostly "classical" geometric constructions, avoiding measurement units and using just a compass and a 90 degree angle ruler, which becomes a very enriching practice and experience. Beyond using Sacred and Reocurring Geometry for my personal artwork, I have grown to believe in its powerful and subtle language and teachings being capable of speaking directly to our higher senses, in a sort of primordial tongue.