Embed Size (px)
Citation preview
1
The triangle that shaped a pedestal for Apollo, a cube, and the head of Vitruvian Man
Leslie Greenhill
© 2015
The 3:4:5-proportion triangle is the most renowned triangle in geometry and has been the subject of interest for millennia. The interest goes right back to ancient
Egypt. At least eight pyramids have the proportions in their design, including the second largest structure at Giza, the pyramid of Khephren, which is adjacent to the Great Pyramid (Petrie, p. 202).1
Figure 1: Khephren’s pyramid
Plutarch (ca. 46–120 AD) is noted for his biographic works and essays on philosophy and ethics. He was a priest at Delphi, a place sacred to the god Apollo.
He discusses the divine status of the triangle in On Isis and Osiris in Moralia V. He writes:
One might conjecture that the Egyptians hold in high honour the most beautiful of the triangles, since they liken the nature of the Universe most closely to it, as Plato in the Republic seems to have made use of it in formulating his figure of marriage. This triangle has its upright of three units, its base of four, and its hypotenuse of five, whose power is equal to that of the other two sides. The upright, therefore, may be likened to the male, the base to the female, and the hypotenuse to the child of both, and so Osiris may be regarded as the origin, Isis as the recipient, and Horus as perfected result. (Babbitt, p. 135)
1 Reference sources are detailed at the end of this essay.
2
For Plato’s discussion of the 3:4:5 proportions in the Republic see Lee, p. 299 (or
Stephanus 546, a standard reference indicator for all translations of the Republic). Memorable examples of the triangle’s properties are given in this essay. The 3:4:5
triangle is illustrated below.
Figure 2
● Location A is the right angle. AB measures 3 units.
AC is 4 units.
BC is the hypotenuse of 5 units.
The sum of all the sides is 12 units.
● AC, as seen, measures 4 units; 4 is a square number: 2 x 2.
The sum of AB 3 and BC 5 is 8 units; 8 is a cubic number: 2 x 2 x 2.
The sum of AC 4 and BC 5 is 9 units; 9 is a square number: 3 x 3.
Two lines are added to the 3:4:5 triangle. They could be readily created by folding the triangle.
3
The first addition
D is the middle of BC, the hypotenuse. From D a line is drawn to A.
Figure 3
AD measures 2.5 units. DC measures 2.5 units. BD also measures 2.5 units.
The second addition
A line perpendicular to BC is drawn from A to E. The new triangle AED emerges.
Figure 4
4
Three measures are of interest in figure 4. They are easily established: ■ AB 3 units is the hypotenuse of the new 3:4:5 proportion triangle AEB. The
angle at B has not changed.
Angle AEB is a right angle, like angle BAC. Consequently angle BAE must be the same as the angle at C.
So, the three measures of interest are:
(1) BE is three-fifths of AB 3 units and measures 1.8 units. (2) AE is four-fifths of AB 3 units and measures 2.4 units.
(3) Since BD measures 2.5 units and BE is 1.8 units, then ED must measure 0.7 of
a unit: 2.5 minus 1.8 = 0.7. ED 0.7 is one-tenth the measure of the sum of AB 3 and AC 4 (3 + 4 = 7).
The highlighted triangle AED is a new Pythagorean triple, a special class of right angle
triangles like ABC. Pythagorean triples are right angle triangles with whole number sides and interesting characteristics.
Triangle AED has the proportions 7:24:25.
The measures in AED reflect its proportions: ● ED is 0.7 of a unit
● AE is 2.4 units
● AD is 2.5 units.
5
In a triangle with 7:24:25 proportions (illustrated in figure 5 below), the sum of the two sides connected to the “7” side add to 49: AE 24 + AD 25 = 49 units.
Figure 5
Forty-nine is a square number: 7 times 7. Seven is the square root of 49. ED measures 7
units.
Interesting matters regarding Roman and Greek measures worth contemplating arise: a) The Roman foot was divided into 12 Roman inches (uncia) or 16 Roman digits
(Rowland and Howe, pp. 189 – 192). The Roman foot measured about 296 mm or
approximately 11 2/3 British imperial/US inches.
A Roman inch was 4/3 times the length of a Roman digit. A 4:3 ratio is found in a
3:4:5 triangle.
6
b) A Roman stade distance measure of 625 Roman feet was equal to a Greek stade2 of
600 Greek feet (Rowland and Howe, p. 167, footnote 90). The relevant Greek foot
measure, found in the Parthenon in Athens, was, accordingly, 25/24 times the length of
a Roman foot (Wilson Jones, p. 74). The “Parthenon” foot measured about 308 mm or about 12.15 British imperial/US
inches. Information given by Herodotus in his book The Histories provides supporting evidence for the existence of the 308 mm Greek foot.
The Roman writer Pliny the Elder, who died in the dramatic eruption of Mt. Vesuvius
in 79 AD, is one of the sources for information about the 25:24 proportional
relationship between the Greek foot and the Roman foot.
Put simply, the “Parthenon” Greek foot was equal to 12 ½ Roman inches. The Roman
foot, as noted in point (a) above, contained 12 Roman inches.
The 25:24 ratio as it manifests in a 3:4:5 triangle can be seen in figure 4.
The number 625 mentioned earlier can be expressed as 5 x 5 x 5 x 5, or 25 x 25. Note that a square drawn on hypotenuse AD 25 in the Pythagorean triple illustrated in
figure 5 would contain 625 square units. The Roman stade contained 625 Roman feet.
Furthermore, in the same triple, the product of 25 (AD) multiplied by 24 (AE) is 600. The Greek stade contained 600 Greek feet.
The data in points (a) and (b) above is explored further from radically different perspectives in five new interrelated essays by the present writer. The essays are entitled:
● Leonardo, Vitruvius, Plato and a triangle to remember
● Discovering Plato’s design for Atlantis, his curious number for Man and Vitruvius’s enigmatic formulation for the “well shaped man”
● Uncovering Plato’s design for Atlantis and its link to his curious number for Man
● How Leonardo, Herodotus and a Roman architect dealt with Apollo
● The Great Pyramid through the eyes of Herodotus
The 3:4:5 triangle and the Greek god Apollo
The god Apollo is mentioned numerous times in the renowned treatise The Ten Books on Architecture by the Roman architect Vitruvius, notably in relation to the replacement of a cracked pedestal for a statue of the god. Vitruvius lived in the first century BC. As can be
seen in his book, he was an admirer of Plato and Pythagoras.
2 The English word “stadium” derives from the ancient Greek word for “stade”.
7
Figure 6: statue of Apollo
(image courtesy Wikipedia)
The description of Apollo’s pedestal is given in the last book of the The Ten Books on
Architecture, that is, in Book 10. Vitruvius writes that the pedestal is twelve feet long, eight feet wide and six feet high (Morgan, p. 289/Book 10.2.13).
Figure 7: Representation of Apollo’s pedestal
(measures in feet)
The volume is easily established as being 576 cubic feet: 12 x 8 x 6 = 576. The number 576 is a square number: 24 x 24.
Especially note that the ratio 8:6 on the front face of the illustrated pedestal is the same as 4:3. Consequently, on the faces of the pedestal that have these dimensions (shown
below), the diagonal must measure 10 feet because it is the hypotenuse, the “five” side, of a 3:4:5-proportion triangle. The 3:4:5 triangle has often been linked to Pythagoras and the Pythagoreans—for example, see Morgan, pp. 252–3/Introduction to Book 9.6–7.
8
The sum of 6 + 8 + 10 is 24.
Figure 8: Representation of the pedestal that includes
the ten-foot diagonal measure
It is possible to discern now why the pedestal was made to be twelve feet long: it ensures the volume becomes 576 (24 x 24) cubic feet. This is not the only 24 x 24 formulation in
Vitruvius’s book. Another is detailed in other work by the present writer.
On manifestations of 24
This discussion begins with a return to the 3:4:5 triangle illustrated in figure 4, which is repeated below.
Figure 9 (formerly figure 4)
9
One additional line is added to the geometry. EF is drawn perpendicular to AB. The new
3:4:5-proportion triangle AFE emerges. Note that location F stems from the creation of a continuous line, bent like a lightning flash, drawn from location D: D to A to E to F.
Figure 10: the “lightning flash” DAEF
Features of interest
(a) EF measures 1.44 units: 1.44 is 1.2 squared. (b) The perimeter of triangle AFE measures 5.76 units:
(EF 1.44 + AF 1.92 + AE 2.4 = 5.76 units).
The number 5.76 is 2.4 times 2.4, that is, 2.4 squared. Compare this with the material on the volume of Apollo’s pedestal (576 cubic Roman feet) in the preceding section The 3:4:5 triangle and the Greek god Apollo.
(c) Imaginatively, in triangle AFE, the hypotenuse AE 2.4 units can be said to be the
square root of the triangle’s perimeter of 5.76 units. (d) The sum of EF 1.44 units and BE 1.8 units is 3.24 units: 3.24 is a square number,
1.8 x 1.8.
The cube in the triangle
A cube created from the 3:4:5 triangle’s features provides a notable conclusion to this essay.
Below is the triangle illustrated in figure 9 and figure 4. It is now labelled figure 11.
10
Figure 11
Line AD is deleted from the triangle and this leads to figure 12.
Figure 12
If BE is folded so that it is at right angles to EC, a rectangular solid like that shown in
figure 13 below can be created.
11
Figure 13: the rectangular solid
The volume of this solid is 13.824 (1.8 x 3.2 x 2.4) cubic units. The number 13.824 is a
cubic number: 2.4 x 2.4 x 2.4. Consequently, the volume of the rectangular solid is equal to the volume of a cube that has sides which measure 2.4 units.
Figure 14: the cube
The formulation for the head of Vitruvian Man as specified by Vitruvius
As mentioned earlier, the Roman architect Vitruvius is the author of The Ten Books on Architecture, one of the most influential books in history, particularly during the Italian
Renaissance. Little is known about Vitruvius, who lived in the first century BC, apart from what he tells about himself in his book. In his treatise, Vitruvius provides, amongst
other things, rules and directions for the design of temples, theatres and war machines. A mysterious and much discussed mathematical formulation for a “well shaped man” (Vitruvian Man) is also given (Morgan, pp. 72–4/Book 3.1.1–7). The formulation was
12
adapted by Leonardo da Vinci to create the now famous illustration of Vitruvian Man, the
man in the square and the circle, shown below.
Figure 15: Vitruvian Man as rendered by Leonardo da Vinci
As can be seen in the drawing, the measure of the outstretched arms is the same as the height of the man—hence the square. Vitruvius prescribes this setup. The “well shaped
man”, reports Vitruvius, is six feet tall. The measure, he says, stems from an ancient notion that the ideal man’s height is six times the length of his foot.
The Roman foot, like the Greek foot, had 16 digit divisions. The Roman cubit, also like the Greek cubit, had 24 digit divisions (Dilke, p. 26 or Rowland and Howe, pp. 189 –
192). Thus, Vitruvius plainly establishes that the “well shaped man” is 96 digits tall (96 digits =
6 feet = 4 cubits).
All the above enables an in-depth examination of the head and face of Vitruvian Man to be made.
13
Vitruvius details the following features (Morgan, p. 72/Book 3.1.2): ● The head from the chin to the crown (vertically) is an eighth of the body height.
An eighth is 12 digits.
● The face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height. A tenth is 9.6 digits.
● The face is divided into three equal parts. Each part therefore measures 3.2 digits. The head measures 12 digits and the face measures 9.6 digits so the distance from the
lowest roots of the hair to the crown must measure 2.4 digits. All this is illustrated in figure 16 below.
Figure 16
The geometry that appeared as figure 12 earlier is shown below (slightly modified) as figure 17.
14
Figure 17: Measures in units
● The perimeter of triangle ABC measures 12 units.
(AB 3 + AC 4 + BC 5 = 12) The head measures 12 digits.
● The perimeter of triangle AEC measures 9.6 units. (AE 2.4 + EC 3.2 + AC 4 = 9.6)
The face measures 9.6 digits. ● EC measures 3.2 units.
The face has three equal parts. Each part measures 3.2 digits.
● AE measures 2.4 units. The distance from the lowest roots of the hair to the crown measures 2.4 digits.
The illustration of the head and face of Vitruvian Man shown above strictly adheres to Vitruvius’s formulation. That it completely accords with the features found in the 3:4:5 triangle is an especially notable fact. This material acts as a precursor to forthcoming
works that demonstrate how and why the 3:4:5 triangle influenced the composition of the characteristics of Vitruvian Man.
Closing remarks
This paper should be seen as a prelude to the five essays listed earlier. Data in the first
listed work alone, Leonardo, Vitruvius, Plato and a triangle to remember, changes the history of the classical world and the Renaissance both rapidly and incontrovertibly.
15
References Babbitt, Frank Cole. Plutarch: Moralia. Volume V. Harvard University Press, 1999 edition.
Dilke , O. A. W. (Reading the Past series) Mathematics and Measurement. British Museum Publications, London, 1991. Lee , D. (H. D. P) Plato: The Republic. Penguin, England, 1987 edition.
Morgan, M. H. Vitruvius: The Ten Books on Architecture . Harvard University Press, Cambridge, Mass., 1914. (Republished by Dover Publications) Petrie , W. M. F. The Pyramids and Temples of Gizeh. Field and Tuer (Ye Leadenhalle Presse), London, 1883.
Rowland, I. D. and Howe , T. N. Vitruvius: Ten Books on Architecture. Cambridge University Press, 1999. Wilson Jones , M. “Doric measure and architectural design 1: The evidence of the relief from Salamis” American Journal of Archaeology. 104 (January 2000) pp. 73–93, Archaeological Institute of America
Credits/Acknowledgements
(All images for educational and study purposes only.)
● Drawing of Vitruvian Man by Leonardo da Vinci, Galleria dell Accademia, Venice
Author’s email address
or
150715 Sevens 24 VM
