numeracy through Geometry and astronomy
Below are lessons in what are thought to have been megalithic methods, that involved measures, geometrical procedures and astronomical techniques.
- Geometry 1: π
understanding the megalithic: circular structures: part 1 It would require 3 and a bit diameters to wrap around the circle – the ratio of 3 and a bit diameters to the perimeter is known as “Pior π: The constant ratio of a circle’s circumference to its diameter, approximately equal to 3.14159, in ancient times approximated … Continue reading “Understanding megalithic methods” - Geometry 2: Maintaining integers using fractionsunderstanding the megalithic: circular structures: part 2 The megalithic sought integer lengths because they lacked the arithmetic of later millennia. So how did they deal with numbers? There is plenty of evidence in their early monuments that today’s inch and foot already existed and that these, and other units of measure, were used to count … Continue reading “Geometry 2: Maintaining integers using fractions”
- Geometry 3: Making a circle from a counted length
The number of days in four years is a whole number of 1461 days if one approximates the solar yearFrom Earth: the time in which the sun moves once around the ZodiacThe 12 constellations through which the sun passes in the solar year of 365.2422 days, now known to be caused by the orbital period of the Earth around the Sun. to 365¼ days. This number is found across … Continue reading “Geometry 3: Making a circle from a counted length” - Geometry 4: Right Triangles within Circles
This lesson is a necessary prequel to the next lesson. It is an initially strange fact that all the possible right triangles will fit within a half circle when the hypotenuse equals the half-circles diameter. The right angle will then exactly touch the circumference. From this we can see visually that the trigonometrical relationships, normally … Continue reading “Geometry 4: Right Triangles within Circles” - Geometry 5: Easy application of numerical ratios
above: Le Manio Quadrilateral The last lesson showed how right triangles are at home within circles, having a diameter equal to their longest side whereupon their right angle sits upon the circumference. The two shorter sides sit upon either end of the diameter (Fig. 1a). Another approach (Fig. 1b) is to make the next longest … Continue reading “Geometry 5: Easy application of numerical ratios” - Astronomy 1: Knowing North and the Circumpolar Sky
about how the cardinal directions of north, south, east and west were determined, from Sacred Number and the Lords of Time, chapter 4, pages 84-86. Away from the tropics there is always a circle of the sky whose circumpolar stars never set and that can be used for observational astronomy. As latitude increases the pole … Continue reading “Astronomy 1: Knowing North and the Circumpolar Sky” - Astronomy 2: The Chariot with One Wheel
What really happens when Earth turns? The rotation of Earth describes periods that are measured in days. The solar yearFrom Earth: the time in which the sun moves once around the Zodiac, now known to be caused by the orbital period of the Earth around the Sun. is 365.242 days long, the lunation period 29.53 … Continue reading “Astronomy 2: The Chariot with One Wheel” - Geometry 6: the Geometrical AMY
By 2016 it was already obvious that the lunar month (in days) and the PMYproto-megalithic yard of 32.625 (261/8) day-inches, generated at Le Manio Quadrilateral as the difference between three solar and three lunar year counts., AMYA megalithic yardAny unit of length 2.7-2.73 feet long, after Alexander Thom discovered 2.72 ft and 2.722 ft as units within the geometry within the megalithic monuments of Britain and Brittany. which, in inches, expresses the true astronomical ratio of mean solar months to lunar months. and … Continue reading “Geometry 6: the Geometrical AMY” - Geometry 7: Geometrical Expansion
above: the dolmenA chamber made of vertical megalithsStructures built out of large little-altered stones in the new stone age or neolithic between 5,000-2,500 (bronze age), in the pursuit of astronomical knowledge. upon which a roof or ceiling slab was balanced. of Pentre Ifan (wiki tab) In previous lessons, fixed lengths have been divided into any number of equal parts, to serve the notion of integer fractions in which the same length can then be reinterpreted as to its … Continue reading “Geometry 7: Geometrical Expansion” - Astronomy 3: Understanding Time Cycles
above: a 21-petal object in the Heraklion Museum which could represent the 21 seven-day weeks in the 399 days of the Jupiter synod. [2004, Richard Heath] One of the unfortunate aspects of adopting the number 360 for calibrating the EclipticThe path of the Sun through the sky along which eclipses of sun and moon can … Continue reading “Astronomy 3: Understanding Time Cycles” - The Fourfold Nature of Sun and Moon
A previous post explained the anatomy of the primary celestial cycles of the Sun and Moon. The “resting” part of these cycles are the winter solsticeThe extreme points of sunrise and sunset in the year. In midwinter the sun is to the south of the celestial equator (the reverse in the southern hemisphere) and in … Continue reading “The Fourfold Nature of Sun and Moon” - Vectors in Prehistory 2
In early education of applied mathematics, there was a simple introduction to vector addition: It was observed that a distance and direction travelled followed by another (different) distance and direction, shown as a diagram as if on a map, as directly connected, revealed a different distance “as the crow would fly” and the direction from … Continue reading “Vectors in Prehistory 2” - Working with Prime Numbers
Wikipedia diagram by David Eppstein : This is an updated text from 2002, called “Finding the Perfect Ruler” Any number with limited “significant digits” can be and should be expressed as a product of positive and negative powers of the prime numbers that make it up. For example, 23.413 and 234130 can both be expressed as … Continue reading “Working with Prime Numbers” - Double Square and the Golden Rectangle
above: Dan Palmateer wrote of this, “it just hit me that the conjunction of the circle to the golden rectangleA rectangle whose sides are in the ratio of the Golden MeanThe Golden Mean is that unique ratio {1.618034}, relative to ONE {1}, in which its square and reciprocal share the same fractional part {.618034}. It is associated with the synodic period of the planet Venus, which is 8/5 {1.6} of the practical year {365 days}, by approximation. It is a key proportion found in Greco-Roman and later "classical" architecture, and commonly encountered in the forms living bodies take. (1.618034) or a Fibonacci approximation to the Golden Mean. existed.” Here we will continue in the mode of a lesson in Geometry where what … Continue reading “Double SquareA unit rectangle of 1 by 2, with important use for alignment (Carnac), cosmology (Egypt) and tuning theory (Honnecourt Man). and the Golden Rectangle” - St Peter’s Basilica: A Golden Rectangle Extension to a Square
above: The Basilica plan at some stage gained a front extension using a golden rectangleA rectangle whose sides are in the ratio of the Golden Mean (1.618034) or a Fibonacci approximation to the Golden Mean.. below: Later Plan for St. Peter’s 16th–17th century. Anonymous. Metropolitan Museum. - The Stonehenge Crop Circle of 2002
One sees most clearly how a single concrete measure such as 58 feet can take the meaning of the design into the numbers required to create it. However, metrology of feet and types of feet can hide the elegance of a design. - The Cellular World of Twelve
The foot has twelve inches just as the British shilling had twelve pence. A good case can be made for twelve as a base like 10, since there are 12 months within the year and many ancient monuments can be seen to have employed duodecimal alongside decimal number, to good effect. - Modularity of Seven
Sacred Geometry is metrical, it is based upon the interactive properties of “natural” (that is, whole) numbers and cosmic constants.