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Re: Designing With the Daisy Wheel #24060 07/20/10 03:01 AM
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D L Bahler Offline OP
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Re: Designing With the Daisy Wheel #24062 07/20/10 10:27 PM
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D L Bahler Offline OP
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Geometry of the Hagia Sophia, Constantinople

The current dome was built when the original collapsed after an earthquake, and its design was based on the geometry of the building it was built upon. This shows Ad Quadratum geometry

The Ad Quatratum system:





Ad Quadratum was apparently considered to be more sacred than Ad Triangulum because it yielded the 8-pointed 'star of Christ' instead of the 6-pointed star of Ad Triangulum. Sacred Geometry (that is, the geometry of Cathedrals and other sacred spaces) seems to have employed Ad Quad. more than ad Triang. The daisy wheel can be considered a form of Ad Triangulum, as it employs the same principles and yield the same geometry.


Even the Pagan-built Pantheon in Rome employs this ancient Euclidean Geometry

Last edited by D L Bahler; 07/20/10 10:36 PM.

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Re: Designing With the Daisy Wheel #24063 07/20/10 10:50 PM
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"There will be those who question the need to begin every architectural geometric design with a circle. There are two cogent reasons. The first is that the architect is aiming in his design to bring Heaven down to earth and Heaven’s geometry is circular. The second reason is even more cogent. It is irrefutable. The fact is that it is impossible to construct a true square, or octagon or octagram or hexagon or hexagram without a pre-existing circle. Even an equilateral triangle requires the intersection of two arcs that is simply in essence the intersection of two circles."


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Re: Designing With the Daisy Wheel #24067 07/21/10 12:40 AM
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Let it be granted:

i. That a straight line may be drawn from any point to any other point.

ii. That a terminated line may be produced to any length in a straight line.

iii. That a circle may be described from any centre, at any distance from that centre.

Euclid's first 3 Postulates

A new point is determined in Euclidian geometry exclusively in one of the following ways:

Having given four points A,B,C,D, not all incident on the same straight line, then:

Whenever a point P exists which is incident on both (A,B) and on (C,D) that point is regarded as determinate.

1. Whenever a point P exists which is incident both on the straight line (A,B) and on the circle C(D) that point is regarded as determinate.

2. Whenever a point P exists which is incident on both the circles A(B), C(D), that point is regarded as determinate. The cardinal points of any figure determined by a Euclidian construction are always found by means of a finite number of successive applications of some or all of these rules 1, 2, 3.

E.W. Hobson Squaring the Circle. History of the Problem(1913, 7-8)

The term Euclidean construction is used for any construction, whether contained within his works or not, which can be carried out with Euclid’s two operations repeated any finite number of times (Hudson 1915, 1ff.). -These two operations are the drawing of straight lines and circles according to his postulates


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Re: Designing With the Daisy Wheel [Re: D L Bahler] #24071 07/21/10 04:14 PM
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Hi D.L.,

OK - now here's the real thing :-

Daisy Wheel

please explain that to me !

Regards

Ken Hume


Looking back to see the way ahead !
Re: Designing With the Daisy Wheel #24074 07/22/10 02:31 AM
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D L Bahler Offline OP
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Ken, What and where is that? And what's with all the scratches all over the place?
The figure in the middle seems to be a simple construction according to the Ad Triangulum or Daisy Wheel mode. The outer circle's radius is equal to the diameter of the inner circle, and two arcs of equal size are scribed on it marking 4 of the six points of the daisy wheel. All the scratches make it difficult to make out.

What is this thing? I have no Idea. Maybe it's the work of some geometer musing with his compass, maybe it's a geometric symbol set into the building for spiritual reasons.

Ever think that the use of the daisy wheel could simultaneously have both of its theoretical applications? Even in a single instance?
The daisy wheel was a tool or at least a form of a tool for geometric design, because without the circle you really can't do anything else. At the same time it was a symbol of protection, harmony, or some type of divine blessing. How so? Because to the Medieval mind, geometry was sacred. Geometry was a reflection of the divine, and of the order of Heaven, particularly since Heaven was thought to be circular in nature, with its center at the Godhead.
The Ad Quadratum reflected the sphere of the earth, with the square pointing off to the four corners of the world.
The Ad Triangulum reflected the elemental essence of creation, with the two triangles overlapping to reflect the full alchemical cycle. Plato is said to have had above the door to his academy a sign that read "let no man that is ignorant in the ways of geometry enter here." or some such. Geometry was the foundation of the ancient world and its philosophy, so why would geometric symbols not hold sacred value to these people?

Don't discount the divine...



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Re: Designing With the Daisy Wheel #24080 07/24/10 10:04 AM
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Hi DL,

Sometimes there are no immediate answers to questions posed and all that could be said is therefore pure conjecture which is certainly not the same thing as hard evidence based fact.

The daisy wheel concerned is on an elm fireplace mantle beam in an Oxfordshire cottage and it might date from the early 1600's. It is not unusual to find such marks but the meaning of same is still not fully understood. Timothy Easton and Laurie Smith are researching these and other types of apotropaic marks.

Regards

Ken Hume

Last edited by Ken Hume; 07/24/10 10:06 AM.

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Re: Designing With the Daisy Wheel #24117 08/06/10 02:57 AM
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D L Bahler Offline OP
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It would be very interesting to hear their conclusions on these things!

I have come to the conclusion that Ad Triangulum/Daisy Wheel geometry is quite a bit easier to use than Ad Quadratum geometry. I suspect that Ad Triangulum or some form thereof may have been the inspiration for scribing daisy wheels all over. The construction of the daisy wheel is, after all, the first part of any Triangulum construction.

I have found that since researching Ad Triangulum and Ad Quadratum, my geometric constructions have become a lot more line oriented. That is, I use lines a lot more than circles any more to establish points.

With these kinds of Euclidean constructions, you also make use of circles of different radii, whereas with a purely daisy wheel construction you just use a single radius, at least at the start. It seems rather confusing, but once you try it things make sense a lot more.

The advantage of Ad Triangulum and Ad Quadratum is also that there is actually some period writing on their use, including instructional manuscripts. Many of these were written when the geometric Gothic architecture began to give way to Classical Revival and other such styles of building and new tools came into play such as the square and the divided rule. These manuscripts were written as an attempt to preserve the then dying (and now dead) High and Liberal Art of Architectural Geometry


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Re: Designing With the Daisy Wheel #24118 08/06/10 09:09 PM
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http://books.google.com/books?id=uSHqAAAAMAAJ&printsec=frontcover&dq=B%C3%BCchlein+von+der+Fialen+Gerechtigkeit&source=bl&ots=_FABRmQscv&sig=8Tp639UHFMaAKZkOxYah4RuxdX4&hl=en&ei=YXdcTMmOGcKnsQbbtvGPBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBIQ6AEwAA#v=onepage&q&f=false

this might be useful to some of you. It's in German, Fraktur script at that, so you can't all read it I know, but it does have many useful diagrams. This book was printed in 1486 in Germany by a master builder named Matthäus Roritzer, and it is about Ad Quadratum geometry. It is an instructional manual on how to use this system


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Re: Designing With the Daisy Wheel #25100 01/09/11 05:14 PM
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I thought I would share this video about the vesica piscis.

http://www.gogeometry.com/videos/sacred_geometry_vesica_piscis.htm

Enjoy;
Jim


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