"The golden and Fibonacci ratios are fascinating ratios that can be represented as spirals, as rectangles, as number sequences and as 3D forms." Burrows, 3D Thinking in Design and Architecture, p. 141.
"The idea of ‘divine proportion’ stems from the mathematician Lucia Pacioli and his three-volume treatise De Divina proportione (1509), which was illustrated by Leonardo. The divine proportion was the golden ratio, or golden section (sectio aurea) as Leonardo called it. It was the ‘mean and extreme’ ratio of Euclid . . ." Burrows, 3D Thinking in Design and Architecture, p. 212.
"Pacioli’s ideas regarding proportion certainly influenced Renaissance artists, including Leonardo, but the De divina proportione itself is short of any new mathematical insights and construction methods, leaving readers in the dark as to how to develop the ratio in art and architecture. It is therefore highly debatable whether such artists actually used the golden ratio in any mathematically precise sense, but as an aesthetically satisfying visual proportion it is found in innumerable Renaissance works of art." Burrows, 3D Thinking in Design and Architecture, p. 212.
Describe regular polygons. The seven-sided regular polygon, the heptagon, cannot be constructed with a compass and straightedge. Instructions for approximating awith just these tools were printed in Geometria Deutsch, a fifteenth-century booklet on practical geometry.
NOTE: There are "exact constructions" and "approximate constructions".
As its name indicates, an equilateral triangle has three sides of equal length. To construct one, draw a horizontal line the length of a side. Place the leg of a compass on one of the line's end points and the compass point on the other end point. Draw an arc over the line and past its midpoint. Do the same from the line's other end point. Finally, draw a line from each end point to the point where the arcs intersect.
This is a standard way to construct a square. Begin by drawing a line the length of a side of the square. Mark the end points a and b, then extend the line beyond b. Place a compass leg on b, extend the compass arm to any point on the line ab, and mark point c. Then, without changing the arm extension, mark point d on the other side of b. Move the compass leg to c and draw and arc of any radius above and beyond b. From point d, draw an arc with the same radius; the two arcs intersect at point e. Next, draw a line from b through e and beyond e – this line is perpendicular to the line ab. To find the top right corner of the square (point f), place the compass leg on b, extend the arm to a, and draw an arc through the perpendicular line. The top left corner of the square (point g) is where two arcs, also with radius ab, intersect, one centered at f and the other at a.
This is the standard way to construct a pentagon. Begin by drawing a circle, then bisect it vertically and horizontally. Place the leg of a compass on point a, extend the arm to b (the circle's midpoint), and draw an arc to find points c and d on the circle's perimeter. Draw a vertical line from b to c. Point e is where this vertical crosses the circle's horizontal bisection line. Place the compass leg on e and draw an arc from f (the top of the circle) through the horizontal bisection line. Where the arc and the horizontal bisection line intersect is point g. Place the compass leg on f, extend the arm to g, and draw an arc through the circle to find points h and i. Place the compass leg on h, extend the arm to f, and draw an arc through the circle to find j. Then place the compass leg on i, extend the arm to f, and draw an arc through the circle to find k. Finally, draw the pentagon by connecting f and h, h and j, j and k, k and i, and i and f.
Anwas described by Mathes Roriczer in the fifteenth century.
A hexagon is a 6-sided regular polygon. To construct one, draw a circle. Open a compass to the circle's radius by placing the leg on the circle's midpoint and the arm on the circle's perimeter – the length of the hexagon's side is equal to the circle's radius. Starting anywhere on the circle's perimeter, mark off points around the circle that are this distance apart, then draw lines between the points.
This construction appears in Mathes Roriczer's Geometria Deutsch, printed in 1489. It is still used today. Begin by drawing a square. Find the square's midpoint by drawing its diagonals. Place a compass leg on one corner of the square, open it to the square's midpoint, and draw an arc through both sides of the square. Draw identical arcs around the square's remaining three corners. Where an arc crosses the square, mark a point – there are 8 such intersections. The distance between any two adjacent points is the length of the octagon's sides.
To construct a vesica piscis, draw a circle, then place the compass leg on the circle's perimeter and draw another circle with the same radius. The shape created by the overlap, shown here in violet, is called a mandorla. Equilateral triangles can be inscribed in the upper and lower halves of a mandorla.
A golden rectangle can be divided into a golden rectangle and a square, to infinity. Its width-to-height ratio is 1 : 1.618...1 To draw a golden rectangle, first construct a square. Place the leg of a compass on the midpoint of one of the sides (a). Place the compass arm on an opposite corner of the square (b), and draw an arc. Extend the side around which the arc was drawn until it intersects with the arc. This intersection point (c) establishes the height of the golden rectangle.
A golden spiral is a line that curves exponentially at the rate of 1:φ (1:1.618..., the golden ratio). To draw it, divide a golden rectangle into a square and a golden rectangle. Then divide that golden rectangle into a square and a golden rectangle. Repeat for as many times you desire. Then draw a line through this spiraling construction that becomes smaller (or larger) at a constant rate. This is done by placing the needle point of the compass at the inside corner of each square and drawing an arc through it. (In other words, you have to keep moving the needle point.)
To construct a golden triangle, first construct a golden rectangle. Place the needle point of the compass at one corner of one of the rectangle's longer sides. Place the pencil lead on the other corner of that side and draw an arc. Repeat on the other longer side. Then draw lines from the point where the two arcs intersect to the corners of the opposite short side. (You are essentially removing one of the rectangle's short sides and turning it into a three-sided figure. The lengths of the sides remain the same.)
Root rectangles and dynamic symmetry.
Description of the construction of a root 2 rectangle, from Hiscock, The Symbol at Your Door, p. 205: ". . . the 1 : √2 ratio and various permutations of it have been found in a large number of buildings, often with surprising precision. One possible explanation for this has been suggested elsewhere and derives from the actual procedure for constructing the rectangle. This starts with a square and, with compass point on one corner and radius equal to the diagonal, an arc is swung from it to extend the side of the square to form the side of the rectangle."
NOTE: The HTML code for the square root symbol is & # 8 7 3 0 ;
"A root rectangle is a rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as √2, √3, etc. The root-2 rectangle is constructed by extending two opposite sides of a square to the length of the square's diagonal. The root-3 rectangle is constructed by extending the two longer sides of a root-2 rectangle to the length of the root-2 rectangle's diagonal. Each successive root rectangle is produced by extending a root rectangle's longer sides to equal the length of that rectangle's diagonal." Wikipedia, "Dynamic rectangle."
SEE Fernie, "The Ground Plan of Norwich Cathedral and the Square Root of Two", 1997.
"This name was coined by the Danish Engineer Tons Brunés, in his book The Secrets of Ancient Geometry and Its Use. In that book he claims the sacred cut is found in the layout of many ancient building, including the Parthenon. . . Brunés calls this construction sacred because it contains both square and circle, uniting the earthly and the divine as in the Vitruvian man. Furthermore, it squares the circle. The length of the four arcs equal the four diagonals of the half-square. And, as mentioned, it gives the, the shape universally used for baptistries and baptismal fonts." (Calter, “Ad Quadratum, the Sacred Cut, and Roman Archtitecture,”, but cite Brunés.)
"There is evidence that the Sacred Cut has been used as a foundation for architectural designs. Brunés myriad examples include the Great Pyramid of Khufu, the Parthenon, and the Pantheon (Brunés 1967: vol 1, 123-147; vol 1, 301-310; vol 2, 38-56); more recent analyses involve the layout of a Roman housing complex in Ostia, and the Baptistery of San Giovanni in Florence (Watts and Watts 1986; Williams 1994)." (Wassel, “Art and Mathematics Before the Quattrocento: A Context for Understanding Renaissance Architecture,” 2015: 68.)
In the construction, there are two proportionally related squares. In the extension, there is an additional proportional square – the outer square – established by the outermost intersections of the circles and the diagonals.
The construction of the sacred cut is done with a straightedge and compass as follows: Draw a square1 and its diagonals, then place the compass leg on each of the square's four corners and draw an arc through its center. Draw lines through the square where the arcs intersect with its sides – two horizontal and two vertical. The small square that is thereby created in the center (shown here in pink) is called the sacred cut square.
If the side length of the original (or reference) square is 1, then the side length of the sacred cut square is √2 - 1, or approximately .414. "The construction can be extended inward, by repeating the construction on the sacred cut square. It can also be extended outward [the figure on the right], joining the intersections of the circles and the diagonals, to form a square of which the original square is the sacred cut square." (Calter, “Ad Quadratum, the Sacred Cut, and Roman Archtitecture.”) By means of these extensions and subdivisions, a series of proportionally related squares can be created.
"The Sacred Cut is perhaps the first component in the long history of using ad quadratum relationships in architectural design." (Wassel, “Art and Mathematics Before the Quattrocento: A Context for Understanding Renaissance Architecture,” 2015: 68.) Unless the Egyptians were using ad quadratum proportions – check on this.
1 1.618... is an irrational number – that is, the decimal portion goes on forever with no repeating pattern of numbers. Like all irrational numbers, it cannot be written as a fraction.
2 Note 2.