The figure at the top, in the above book plate shows two tetrahedrons fused together. With one of the vertices facing the viewer's eyes. A motivation for this blog originally was the seemly appearance of scientific advances without experiment to back them At some point a hypothesis needs to be considered. If so, then how can it be devised? Especially if there are no appropriate models to study. This would definitely be the case in the modern field of astronomy. We cannot study a star and planet on a laboratory bench. We are forced to devise models. If the subject has been studied many times this may not seem to be a problem. However, assumptions by many investigators, over time, can also led to the eventually discovery that the initial hypothesis was incorrect. This can be demonstrated to skeptics if new data can be obtain that does not fit the facts. How often though do skeptics following research closely? They are concerned about popular culture. This means art, psychology, and scientific concepts that have been around in the public mind, introduced by media concerns. The mantra of the skeptics is the presentation of a concept in a peer review journal. Pop culture was never presented in a peer review journal. Many concepts of the European Renaissance were not run pass a peer review journal. The origin of the Renaissance came from the East in the form of books. This is where the legendary request from Consumo De Medici (1389-1464, the founder of the Medici banker family) that the translation of Plato 's work can wait. First translate the manuscripts on Hermeticism. Today these translations are called the Corpus Hermeticum, Latin translation from a Greek text by Marsilo Ficino in 1471. Only a portion had been translated. The complete work was introduced in 1500. The knowledge in this book came from the eastern hemisphere of the world and it changed everything in Europe. Like a spiritual awakening does to an individual. Therefore by saying Plato can wait states the opposite of what is taught in western universities. What exactly is in the Heremetic manuscripts that has a dramatic change on people? They describe a practice involving exercises both internal and external to effect psychological transformations through the active participation of the conscious mind. The titles of the chapters alone make one wonder why the texts are not being actively studied everyday. Topics covered are Matter, Time, The Soul, The Power of Choice, The Whole Economy, The Virgin of the World, The Sermon of Isis to Horus (her son). For more confer this link on the Corpus Hermeticum. Problem of Thought Without an Axiom (The Descartes Nightmare of the Brian in the Glass Flask) What if the initial hypothesis proposed for a scientific problem cannot be described by a scientific method? What if it came from a totally different field? This should not be surprising. The field of Robotics came from Psychology and Electrical Engineering. A warning to university staff should be pointed out that research cannot be controlled by limiting the search in a narrow area. Suppose models of the universe came from say psychology, or agriculture rather than a physics lab. Would the skeptics take them seriously? Thus we find ourselves in an entirely new literature search to justify the scientific method. Once again the artists and psychologists can come to our aid. In 1596, Joannes Kepler published The Secret Of The Universe or in a more flashy title Mysterium Cosmographicum, in Latin. His book explained how planets were arranged in the solar system. His model was based on the use of five nested platonic solids inscribed upon spheres. His rational has perplexed many. Fortunately most scientists are not familiar with the work today. The nagging question is why did he choose such an elaborate model? We tell people simple models are better, and closer to Reality, cough. To begin with what are the Platonic solids and more importantly what are their function? The actual solids are constructed from triangles, squares, and in the case of only one solid (the Dodecahedron), pentagrams. Texts refer to them, but never imply what their use is, or where they come form. Indeed why are the called Platonic? They are described obtusely in a book called Timaeus by Plato. In the book these solids are related to the Tattva's of Fire, Air, Earth, Water, Quintessence or Aether (ether). No pictures were provided. These elements are described in one of the hermetic texts entitled Kore Kosmu in Greek and translates as Virgin of the World. The English translation is as follows: And Isis answer made: Of living things, my son, some are made friends with fire, and some with water, some with air, and some with earth, and some with two or three of these, and some with all. And, on the contrary, again some are made enemies of fire, and some of water, some of earth, and some of air, and some of two of them, and some of three, and some of all. For instance, son, the locust and all flies flee fire; the eagle and the hawk and all high-flying birds flee water; fish, air and earth; the snake avoids the open air. Whereas snakes and all creeping things love earth; all swimming things love water; winged things, air, of which they are the citizens; while those that fly still higher love the fire and have the habitat near it. Not that some of the animals as well do not love fire; for instance salamanders, for they even have their homes in it. It is because one or another of the elements doth form their bodies' outer envelope. Each soul, accordingly, while it is in its body is weighted and constricted by these four. End of the Translation. From the above text we can determine these elements have a lot to do with personality and hence consciousness. Neither text described how the Platonic solids are constructed. How to build Three Dimensions from Two Dimensions? By just considering the circle it is possible to devise the 5 Platonic solids. A circle can be an enlarged representation of a dot. A dot is a representation of a thought formed in our consciousness. Notice a dot can also approximate a pixel. Notice in Georges Seurat's work entitled A Sunday Afternoon on the Island of La Grand Jatte (1884). He used the dot to make images, just like a pixel. Imagine your thoughts is such a dot. Imagine how you come up with your ideas, your dreams, and your plans. Dots of different color provide another dimension of information to a two dimensional landscape. Different dots will produce a picture of a family enjoy time together. Of people being surrounded by nature. This idea of the dot allows for your computer to display paintings on the screen. The symbol of a dot represents your initial idea. Yet look at how this model of consciousness is used to create new inventions. This concept is important to understand how we can develop a three dimensional solid with two dimensional polygons, or even a 1 dimensional dot. The simplest structural element in geometry. To construct our circles we need to devise some rules. Suppose we confine the drawing of two circles such that they overlap. This would mean that the origin of one circle be on the circumference of the other circle. The area shared by these two circles is called the Vesica Pisces (Fish bladder, or Mandorla, which means Almond in Italian). If you were to cut the figure along the radius marked as "a, between center 1 and center 2" of the circle you would have the geometry of a church door. Notice how the church door matches up with half of the vesicle pisces in the example below. The exercise can be repeated several times. This produces what is currently being called in popular culture the Flower of Life. It was not always know by this name. By carefully drawing a series of circles to create a rose of circles we come to an image known as the Seed of Life. This exercise can be performed to create elaborate patterns. After performing this exercise a number of times a figure starts to emerge, in the center of the diagram. Therefore this diagram has simple geometric origin. It also explains how one of the Fruits of the tree life can be used to construct the entire tree. This same Tree Of Life is made popular in texts on the Qabbalah, or Kabbalah. Each of these fruits is drawn at the center of a flower. It is more remarkable when the actually tree of life contains up to seven dimensions even though we can only present at most a three dimensional model. Texts on the Tree of Life often show the two dimensional representation as traced in purple (Violet). If we highlight some of the circles to create a star like pattern called the Fruit of Life we find a way into the construction of the Platonic solids. Notice how the construction of two tetrahedrons can be drawn. This provides an optical illusion and the resulting figure is called Metatron's cube. Metatron is Greek for beyond the throne. Notice how a cube, two tetrahedrons and a center flower pattern as the Seal of Solomon can be presented from a. two dimensional drawing. Shading can provide a much better view of the resulting figure. In this view two cubes can be easily seen. All of the Platonic solids can be constructed from this Fruit of Life figure and are listed below as an example. The center of the circles form the vertices of these solid figures. This is an exercise into perspective drawing. This way of constructing the Platonic solids is far different than described in texts on group theory and crystallography. They convince the students to first study the number of planes, and edges and observed how to arrange the two dimensional plane figures to make the three dimensional solids. In Chemistry and Biology these Platonic solids are observed in complex molecules and as the structural coats of viruses. Consider the card players by Paul Ce'zanne 1892 copy. First note the total number of pipes. There are six. Just like the two triangles in the seal of Solomon. An additional vertex of one of the tetrahedrons would include the viewer's eyes. The other vertex would be behind the painting. If you examine the position of the figures in the drawing and then look at the the image of the metatron's cube you will notice a number of geometric properties. In addition two tetrahedron's can be placed where the center figure resides. A question can be asked why then does including the platonic solids in a work of art makes the work more appealing to the viewer? Do we arrange ourselves in our daily activities around the vertices of these platonic solids? Our pictures balanced when done so? By use of perspective the artist can create the power to direct a viewer's attention. The central figure seated has his legs arranged like a triangle.
By the time Kepler published his work another work on graphics was published. It showed platonic solids in futuristic landscapes not unlike the science-fiction comics of Wally Wood. Further Reading Geometry and Perspective by Lorenz Stoer 1567 in German Book on Perspective by Jean Cousin 1560 in French Mysterium Cosmographicum by Johannes Kepler 1596 in Latin Harmonics Mundi by Johannes Kepler 1619 in Latin Harmonies of the World by Johannes Kepler in English
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