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Quote:
Originally Posted by geomancer
One of the disciplines I have been exploring is alchemy. There is a plate (Emblem 21) from "Atlanta Fugiens" by M. Maeir that has been used in various publications. The standard explanation is that the "magician" is trying to square the circle. This explanation is accepted without question or any logical proof. By questioning this belief and looking at the original statement the real answer is there.
It is in actual fact a question and response scenario (Freemasonry initiation ceremony, role playing gamers (D&D)). The question in this case is:
1. Given a cube of a known surface area, draw me a sphere of equal surface area using geometric means (pencil, straight edge, compass).
2. Given a sphere of known surface area, draw me a cube of equal surface area using geometric means.
The answer is Emblem 21.

Geomancer,
this is fascinating.
Here is a small Emblem XXI I found on the internet:
http://www.culture.huberlin.de/HB/v...ialeAbb13.jpg
I knew the image, but the name was not enough to make me remember it.
The internet image is a figure from what seems to be an interesting page about "Die Hermetische Ikonologie der vier Elemente"....it seems to be in German
http://www.culture.huberlin.de/HB/v...konologie.html
Marco




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#11

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Join Date: 28 May 2004
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Marco, the following site will give you some valuable information on Emblem XXI.
http://hdelboy.club.fr/atalanta_fugiens.html (I'm not sure how to make this a live link)
Emblem XXI is from "Atlanta Fugiens" by Michael Maier, a 17th Century alchemist. Be aware, some publications quote a different manuscript.
After studying the manuscript, this is the only image that I found which displays circle/square geometry.
I will endeavour to post some information on della Francesca on the link provided in the next day or so.




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#12

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Emblem XXI
I had a look at the page about Atalanta Fugiens: quite interesting!
It is a long page, and I only looked for the information about Emblem XXI.
It contains a better image of the emblem: http://hdelboy.club.fr/atal21.jpg
It also presents the original text associated to the image:
Quote:
Originally Posted by Godwin, Michael Maier’s Atalanta Fugiens, p. 147
"Make a circle around man and woman, then a square, now a triangle; make a circle, and you will have the Philosopher’s Stone."
The epigram reads,
"Around the man and woman draw a ring,
From which an equalsided square springs forth.
From this derive a triangle, which should touch
The sphere on every side: and then the Stone
Will have arisen. If this is not clear, Then learn Geometry, and know it all."

Very interesting indeed. The text corresponds to the image, but not perfectly. "A triangle, which should touch the sphere on every side"...but in the emblem the triangle does not touch any of the two circles in three points. Emblem XXI is still unclear to me, but I think it is fascinating.
Marco




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#13


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Join Date: 07 Jul 2003
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Geomancer and Marco,
thank you for these puzzles. I love geometry, much more than algebra.
Geomancer's first puzzle I have not been able to figure out yet. I am not sure how to prove the identity of surface area on a sphere against a cube, using only geometric means. I always must resort to formulae, which is of course not the point at all.
The Maier "Philosopher's Stone" is easy to do. But I don't feel that by doing it, I have the Philosopher's Stone. What am I missing?
Perhaps the clue is in the first instruction  draw a circle around a "man and a woman". What does this mean? Why does it have to be around a man and woman  why not just any circle? And if they are important, what exactly do they look like? Are they joined, or apart? How far apart? Are they equal height? Why would any of these things be important, if only geometric means are used?
I think "man and woman" is some kind of alchemical code in this case, not to be taken literally. In any case, whatever might be inside the circle, the geometry can be done independently of it.
There is a second enigma in Maier's text  why does he go from a circle to a sphere? Or is he using "sphere" loosely, to mean the same thing as a circle?
The smallest triangle I can make which solves this puzzle is an equilateral triangle, where one side of the square coincides with one side of the triangle. But other triangles are also possible in this description.




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#14

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Join Date: 28 May 2004
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Ross and other geometers
Thank you for your uplifting post. I was beginning to think nobody else was interested in geometric puzzles.
From my point of view I always use maths/trigonometry in conjunction with pure geometry. This way I can work towards a geometric equation that satisfies the predicted answer.
When you are looking at the esoteric arts, it pays to keep in mind that in times past, there were always two streams, operative and speculative.
Operative being the practical aspect eg. the stone mason, working with materials on the job, and Speculative eg. the architect or engineer, working on the plans for the craftsman to build.
The architect or engineer can work out to the nth degree for a precise fit, but the craftsman knows the physical constraints of precision eg. how much "play" for practical purposes.
With this in mind there will generally be a slight deviation in geometry and the mathematical proof. That is why it is impossible to proove the squareing of the circle to the nth degree of accuracy, but it can be done within the practical accuracy by geometry. Has speculative overtaken reason, the letter of the law vs intent of the law makers?
As for Emblem XXI, it is a step in a series of problems of increasing difficulty with relation to circle/square geometry. This particulr problem takes us from a 2d shape, circle and square to a 3D shape, sphere/cube. The same template is used to solve a number of these problems. In this case the problem is proposed from an alchemy view point (rather cryptic to say the least). The template was recovered from my work on the ViscontiSforza deck.
The male/female represent the natural starting point of the sequence (the first chakra). The problem is aimed at the sixth level (chakra) by my method. It involves the surface area (a 2D problem) of a 3D shape which is associated with volume.
The triangle generated in Emblem XXI is very interesting in that it is of the ration 1: 1+root2. You can draw it (geometrically), but cannot calculate the hypotenuse without introducing the error inherent in root 2. Speculative vs operative?
I hope I have been able to encourage without instilling too much confusion.




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#15

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where? when?
hey guys!
i am very interesting about the geometry in Tarot especially in Tarot de Marseille one so i would like to know if you did publish, post your info or when will you do it?
Thanks!




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#16

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Paolo, Piero and the Ace of Cups
Quote:
Originally Posted by Lolo_Fada
i am very interesting about the geometry in Tarot especially in Tarot de Marseille one so i would like to know if you did publish, post your info or when will you do it?

I am also eager to read more about this topic!
I think http://www.tarotpedia.com/ could be a good place to publish something systematic and structured.
BTW, on a book about Paolo Uccello I found reference to the attached drawing (in the Uffizi Museum, Florence, N. 1758 A). It is generally attributed to Paolo, but in 1964 an historian named Parronchi noticed this passage in Giorgio Vasari's life of Piero:
Quote:
Originally Posted by Giorgio Vasari
In his youth Piero studied mathematics, and even if at the age of fifteen he was bound to be a painter, he never abandoned his studies. He produced wonderful fruits in both mathematics and painting. Therefore, Giudobaldo Feltro, ancient Duke of Urbino, had him as his painter; Piero made for him many beautiful pictures with small figures, but most of them have been lost, since that state has been troubled many times by the war.
Nevertheless some of his writings about geometry and perspective have been preserved. In such matters he was not inferior to anyone in his time, nor possibly in any other time. This is proved by all his works, full of perspective, and in particular by a vase, drawn by means of squares and faces, in such a way that you can see it from before, behind and from the sides, the bottom and the top: it surely is a wonderful thing, since he represented every small detail, and managed to draw all those circles with much grace.

I think Parronchi is right, and this work is one of Piero's geometrical masterpieces. Still Paolo Uccello also is a great master and deserves attention.




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#17

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I presume the quote in the previous post relates to Piero della Francesca. I f so I can confirm his exceptional insight into geometry and perspective. It was in fact one of Piero's manuscripts that Leonardo da Vinci based his studies into perspective on.
Piero has interseted me for some time now. I first came across him as one of the possible artists that that made the extra cards for the ViscontiSforza deck. I have applied my template to some of his paintings and found them to conform to the same underlying geometry that exists within the extra cards as well as the rest of the deck. This has confirmed to me that the second artist (Piero?), had the same knowledge of esoteric geometry as the first artist.
My problem now is that the few examples of Bonifacio Bembo (the attributed first artist) that I have found do not correlate to the template.
The question now arises who could have painted the cards? Logic leads me to reason that the first artist would be of a generation earlier than Piero and have a highly developed knowledge of esoteric geometry to perform the initial grid.
The picture that the ViscontiSforza deck displays when assembled to the pattern is one of a Cathar/gnostic journey/theology. A question now arises, did the first artist (Bembo?) work to a grid supplied by a skilled (Cathar) geometer or did he create it all himself and Piero picked up the pieces so to speak?




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#18

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Quote:
Originally Posted by geomancer
I presume the quote in the previous post relates to Piero della Francesca. I f so I can confirm his exceptional insight into geometry and perspective. It was in fact one of Piero's manuscripts that Leonardo da Vinci based his studies into perspective on.

Yes Geomancer, Vasari is referring to Piero della Francesca.
Quote:
Originally Posted by geomancer
The question now arises who could have painted the cards? Logic leads me to reason that the first artist would be of a generation earlier than Piero and have a highly developed knowledge of esoteric geometry to perform the initial grid.
The picture that the ViscontiSforza deck displays when assembled to the pattern is one of a Cathar/gnostic journey/theology. A question now arises, did the first artist (Bembo?) work to a grid supplied by a skilled (Cathar) geometer or did he create it all himself and Piero picked up the pieces so to speak?

The difference between Bonifacio Bembo and Piero della Francesca is huge (I am not refferring to Tarot in particular): the first is still a "gothic" artist, with a great attention to the "line", his works are more or less flat, reach in color a shapes, but still 2D. Piero is much more interested in volumes and geometry, he also seems to me to be more interested in realism and less in abstract estetical result.
I would compare Bembo to Mondrian and Piero to Leger.
About the idea of a geometer designing the cards whose final painting was performed by Bembo, I think this is possible. In the 16th Century there often was some theoretician at work before an artist could start painting; maybe this also could happen in the 15th Century (but I am not sure it is documented).
Marco




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#19


