Ropestretcher's Triangle, Phi & Fibonacci
© Lorena Loo
© Lorena Loo
Gyorgy Doczi1 among others have written about the Egyptians use of the 3-4-5 triangle in surveying. As the Nile underwent annual inundation, it was necessary for them to use a reliable and convenient surveying tool.
Such a tool came in the form of a rope with twelve knots spaced equally along its length. By holding the rope at the 5th and 8th knot (from the left) and bringing the ends together, a right-angled triangle was automatically formed. The right angle was the requisite angle for surveying. This triangle formed a Pythagorean triple, wherein three positive integral values a, b, c satisfy the relationship a2 + b2 = c2. In this case we have 32 + 42 = 52. The triangle has become known as the ropestretcher's triangle or Egyptian triangle.
Such a tool came in the form of a rope with twelve knots spaced equally along its length. By holding the rope at the 5th and 8th knot (from the left) and bringing the ends together, a right-angled triangle was automatically formed. The right angle was the requisite angle for surveying. This triangle formed a Pythagorean triple, wherein three positive integral values a, b, c satisfy the relationship a2 + b2 = c2. In this case we have 32 + 42 = 52. The triangle has become known as the ropestretcher's triangle or Egyptian triangle.
© Copyright Lorena Loo
The ropestretcher's triangle or Egyptian triangle. Angle between sides of length 3 and 4 is a right-angled triangle.
On another continent during another era, the 3-4-5 triangle also appeared in the megalithic stone monument known as Stonehenge. For example, the Sarsen archways formed by trilithons which bear a remarkable resemblance to the Greek letter for the value of pi, happen to form 3-4-5 triangles. Astronomer Gerald Hawkins believes Stonehenge was an astronomical calculator which could predict events such as eclipses as well as other lunar and solar alignments.
Sarsen archways at Stonehenge. The Heelstone can be seen through the center of the middle archway. The sun appears at the tip of the Heelstone on the first day of summer.
Another place where the 3-4-5 triangle turns up is at the Great Pyramid which will be explored in an article on The Great Pyramid.
Right now we are going to see how the 3-4-5 triangle curiously ties into the golden ratio and the Fibonacci numbers.
Below the 3-4-5 triangle is illustrated. In the second diagram the vertices have been labelled A, B, C and angle ABC (the angle at vertex B) is bisected. The line bisecting this angle intersects side AC at point D. Remember that angle ACB is 90º.
Right now we are going to see how the 3-4-5 triangle curiously ties into the golden ratio and the Fibonacci numbers.
Below the 3-4-5 triangle is illustrated. In the second diagram the vertices have been labelled A, B, C and angle ABC (the angle at vertex B) is bisected. The line bisecting this angle intersects side AC at point D. Remember that angle ACB is 90º.
Next we construct a circle centered at point D with radius DC as illustrated in the diagram to the left below. The circle intersects the line bisecting the angle at B at points E and F as labelled.
In the diagram to the right below, we have constructed the perpendicular line to AB from point D. That perpendicular meets AB at point G.
In the diagram to the right below, we have constructed the perpendicular line to AB from point D. That perpendicular meets AB at point G.
We discover the following relationships/ratios are true:
(Time permitting, I will post my proofs of these relations and link to the page from this article as well as show the square root of 5 also appears. Mathematician H.E. Huntley briefly touches upon this construction and the relations as well in his book The Divine Proportion. But let the reader beware. Huntley's book contains a number of errors and there are two errors with regards to these particular relations. One reason I have a habit of working out mathematical relationships whenever possible is precisely because I sometimes find significant errors.)
By this simple ruler and compass construction on a 3-4-5 triangle, we arrive at ratios of lengths involving the golden ratio and its square as well as the ratio of consecutive Fibonacci numbers. The latter are, of course, approximations to the golden number! Absolutely amazing.
By this simple ruler and compass construction on a 3-4-5 triangle, we arrive at ratios of lengths involving the golden ratio and its square as well as the ratio of consecutive Fibonacci numbers. The latter are, of course, approximations to the golden number! Absolutely amazing.
Text and images (unless otherwise indicated in the credits) are copyrighted
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
Footnotes:
1The Power of Limits: Proportional Harmonics in Nature, Art, and Architecture by Gyorgy Doczi; Shambhala; 1994
Images:
Stonehenge Sarsen archways diagrams from The Power of Limits: Proportional Harmonics in Nature, Art, and Architecture by Gyorgy Doczi; pg 40
Bas relief from tomb of Rameses VI from Serpent In the Sky by John Anthony West; pg 51; Julian Press; 1987 Edition
1The Power of Limits: Proportional Harmonics in Nature, Art, and Architecture by Gyorgy Doczi; Shambhala; 1994
Images:
Stonehenge Sarsen archways diagrams from The Power of Limits: Proportional Harmonics in Nature, Art, and Architecture by Gyorgy Doczi; pg 40
Bas relief from tomb of Rameses VI from Serpent In the Sky by John Anthony West; pg 51; Julian Press; 1987 Edition
Direct evidence of the Egyptians knowledge and implementation of the 3-4-5 triangle appears to be rare, at least according to John Anthony West. A bas relief from the tomb of Rameses VI as illustrated below appears to be one of the rare instances of such evidence.
