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.