Recent postings to the sundial list have pointed out that on the equinox, the path of a shadow cast on a horizontal surface is a straight line oriented west to east. Only on the equinox is the path of any shadow a straight line. When the shadow is cast on other planes, it is still a straight line but the orientation depends on the inclination and declination of the plane. For vertical planes, the angle of the equinox declination line is equal to the substyle difference SD for a vertical declining dial. Tan SD = Sin Dec / Tan Lat.
Earlier this year I posted a note that this mathematics can explain the equinox phenomenon at the Maya temple of Kukulcan at Chichen Itza near Cancun. I have re posted the note below explaining the phenomenon of descending serpents on the equinox. Further analysis indicates that there were several errors in the initial analysis. The dimensions and angles were not corrected for perspective. The orientation was magnetic rather than true. The plane specified was the plane along the wall rather than perpendicular. The original math led me to just the answer I was seeking so I posted the note. Unfortunately that answer was wrong. My enthusiasm got ahead of my analysis, but the analysis is continuing. Stay tuned. The remarkable phenomenon exists. Over 50,000 people witness it each year. However, the more I study it, the more skeptical I become that it was a planned design feature of this particular structure. There is nothing unique about the location, orientation or style of this temple. The equinox phenomenon may be a modern over interpretation of an accidental feature. The pseudo science of pyramid lore is almost as bad as magic crystals and energy vortices. My interest in this subject has led me to some interesting areas. One that I would like to recommend is a 3D virtual reality tour of the temple of Kukulcan. Go to <http://www.halfmoon.org/pyramid.html>. Download the small 14 kb file pyramid.zip. To view it you will need a virtual reality player. As recommended, go to <http://www.virtus.com> and download the free player "Virtus Player 4.0". On the Virtus home page, click on "Downloads". On the "Download" page click on the small print "Use one of our free players". Go to the bottom of the next page and download the appropriate Virtus Player 4.0. This is a 3 MB file that took me 15 minutes to download. Install the player, unzip the Pyramid file and run it with the player. Then take a walk through or orbit around the Maya temple of Kukulan. It does not show the equinox phenomenon but it is an interesting tour. My thanks to Nancy McNelly @ halfmoon for posting the file. Enjoy the tour and the equinox. Let's celebrate the return of the sun to the northern hemisphere, the end of winter and the first day of spring. Roger Bailey Walking Shadow Designs 51 N 115 W The original (corrected) posting follows. "A highlight of my recent trip to Mexico was the Mayan ruins at Chichen Itza, in particular, El Castillo, the Temple of Kukulcan. This great pyramid is designed to mark the yearly cycle of the sun. Each of the four faces represents a season. Each of the four stairs has 91 steps, the number of days in each of the four seasons. The temple at the top represents the 365th day. The winter solstice sun just grazes the north face as the 45 degree angle equals the noon azimuth. (90-lat 20.666 - dec 23.44 - semidiameter .25 = 45.6 degrees. The light and shadow phenomenon of the equinox is world famous. See <http://www.piramideinn.com/equinox.htm>. The corners of the nine platforms cast a series of triangles on the stairway. As the sun sinks, the triangles of light move up creating the illusion of a serpent ascending from the ground to the temple. Tens of thousands come to this site at the spring and fall equinoxes to witness this remarkable phenomenon of the serpent descending to the ground as the sun rises and then ascending in the afternoon as the sun sets. The show takes over 3 hours and peaks near sunset when all nine triangles of light show the whole serpent. I could not find a good technical analysis of the phenomenon. The description in a booklet I bought has serious flaws. Most descriptions have more mysticism that facts. I asked myself if sundial design math could elucidate the phenomenon. Declination lines are the solution! The concept of declination lines is familiar to most of you. These lines are the path of the tip of the gnomon shadow for various dates and solar declinations. When the declination and latitude have the same sign, these lines are hyperbolic curves towards the gnomon. When the latitude and declination signs are contrary the lines curve away. On the equinox, the declination lines are straight lines. This is a universal phenomenon. The path of all shadows cast on any plane surface at any latitude are straight lines. On the horizontal plane, the equinox declination line is due east west. On a south facing vertical plane the line is horizontal. On a vertical declining plane, the declination line is sloped at an angle equal to the Substyle Distance (SD) of a vertical dial on that plane. The usual vertical declining design equation applies. Tan SD = Sin Dec / Tan Lat where Dec is the declination of the vertical plane from south. This special case of the straight declination line on the equinox is the basis of the Kukulcan ascending serpent effect. At Chichen Itza, the latitude is 20.666 and the orientation of the pyramid is 18 degrees off the north south axis. In this case, the formula reduces to SD = 39 degrees. This is exactly the angle the staircase of the tower makes with the horizontal plane. This explains the movement of the shadows along the face of the staircase creating the illusion of the serpent ascending as the sun sets. Each triangle of light and shadow is in effect a separate gnomon casting a shadow moving in a straight line at a 39 degree slope. A remarkable phenomenon, simply explained with the mathematics of sundials. This investigation required a lot of direct experimentation. I had to built several sand castle models on the beaches of Cancun and Isla Mujeres to test the solar orientations. I may now have to build one at home, adjusting for the latitude difference by tilting the model by 30 degrees. This should demonstrate the effect just as well as the original but snow may not be the appropriate construction material. Have any of you witnessed the event? Do you know of other examples of light and shadow shows built into solar oriented structures? Do you know of any good references to more complete technical analyses of the phenomenon at Chichen Itza? Roger Bailey Walking Shadow Designs N 51 W 115"
