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What is a Pyramid?
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Give a plane polygon and a point, which isn't inside the plane of the
point.
If you connect this point with the corners of the polygon, you get a
(general) pyramid. |
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If the polygon is a square and the point lies over the centre of the
square, the straight square pyramid, shortly called pyramid, develops.
The following site relates to this regular pyramid. |
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Pieces of the Pyramid top
Lengths
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The pyramid is determined by the side a of the base and the height
h.
The triangular side s and the triangular height h' are further lines. |
Angles
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The angle between a triangle and the base (angle of inclination) is
typical for the form of the pyramid. |
Surface
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The four lateral faces form M (M=2ah').
The lateral surface O consists of the base and the lateral faces.
(O=a²+2ah').
(German: Mantel M=coat, Oberfläche O=lateral faces) |
Volume
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If you put a rectangular solid with the volume a²h around the
pyramid and move the vertex to the corners of the solid, you get a crooked
pyramid with the same volume. There are two additional pyramids of the
same volume. They all fill the solid.
The volume of one pyramid is V=(1/3)*a²h. |
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Pyramid and Cube top
If you succeed in making the 3D-view, you can three-dimensionally look
at the following three cube pairs.
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If you draw the four space diagonals of a cube, the cube is divided
in six equal pyramids. The height of a pyramid is a/2. There is V=a³/6=a²*(2*h)/6=(1/3)*a²*h.
This again is the well known formula for the volume of a pyramid. |
The angle of inclination is 45°.
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If you connect the centre of one square and the corners of the
opposite square, then a pyramid with the feature h=a develops.
The angle of inclination is arc tan(2)=63.4°. |
Equilateral Pyramid top
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If all edges of a pyramid are the same (a=s) and if you reflect it
at the base square, a double pyramid develops, which is only formed by
equilateral triangles. This solid of eight triangles is called an octahedron.
The angle of inclination of the pyramid with the equilateral triangles
is
arc cos[sqrt(3)/3]= 54,7°.
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The octahedron (the same as the tetrahedron, the cube, the dodecanhedron,
and the isocahedron) belongs to the regular polyhedra or Platonic solids.
You can also produce an octahedron by connecting
the centres of the squares of a cube by lines. Cube and octahedron are
dual.
Pyramidal Numbers top
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You can build a pyramid with layers of spheres. The number of the spheres
in one layer is a square number: 1,4,9,16,... , generally n². If you
add the spheres layer by layer, you get the pyramidal numbers 1,5,14,30,...
, generally 1+4+9+16+...+n²=n(n+1)(2n+1)/6. |
In former times people kept cannon balls in such a way. They could easily
count them by counting the number of the layers.
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If you glue 14 marbles to two groups with six and to one pair, you
get a puzzle: You must form a pyramid with three pieces. |
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This puzzle is more complicated. ........................... |
It looks nice, if you form a pyramid with single marbles. Then the marbles
in the lowest layer must lie in half rounds or in a framework.
The Great Pyramid of Gise top
If you speak about pyramids, you usually mean
the Great Pyramid, Pharao Cheop's tomb, from the 4th dynasty (2500 BC)
situated 15km south of the centre of Cairo in sight of the Nile in Epypt.
Cheop's pyramid is a building with a string of
superlatives: It is one of the Seven Wonders of the World, which is mainly
preserved. It was the largest building up to the modern era. It is one
of the most famous buildings of the world.
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The measurements of Cheop's pyramid differ in the literature. I use
the data of a new travel guide (5) hoping the last researches are considered.
Today the pyramid is 137.0m high and 230.5m long. Originally it was a little
larger (on the left). It covers an area of about 5 ha. About 2,5 millions
blocks with a volume of nearly 1m³ form the pyramid. The covering
of polished lime plates is not preserved. |
If you give the original data a=232.7m and h=146.6m, the edges are s=220.4m,
the base 5.4150 ha, the lateral faces 8.7120 ha, the lateral surface 14.13
ha, the volume 2646000 m³ and the angle of inclination 51.6°.
The volume is illustrated. If you let a stone block be a cube with the
edge 1m, you can form a line 2500km long. This is nearly the distance London
- Athens.
Number Mysticism
of the Great Pyramid top
Statements, Comments
1
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Area of the square above the height:
h² = 21780m².
Area of a triangle side: (1/2)*a*h' = 21490m².
Supposed: The areas are the same.
(Herodot). |
The equations h²=(1/2)*a*h' and h²+a²/4=h'² (Pythagorean
Theorem) lead to the ratio a:h= sqr(sqr(20)-2)=1,5723... . This is about
3.1446../2 or Pi/2.
Both equations result in the second ratio h':(a/2)=1/2*[1+sqr(5)].
This is the golden ratio phi = =1.6180... .
2
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Perimeter of the square base: 4a=930.8m
Perimeter of the circle with the radius h: 2*PI*h=921.1m
Supposed: The perimeters are the same. |
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The ancient Egypts knew the ratio (circle
perimeter : circle diameter) as 256/81 (Rhind Papyrus 1850 BC). This
leads to Pi=3.16... (book 4).
The circle as a symbol for the sun was very
important in the ancient Egyptian mythology and decorates the head of the
Goddess Hathor (on the left) for instance.
The ratio was explained on TV by Hoimar
von Ditfurth (5) this way: A circle is unrolled along a square side of
Cheop's pyramid and the same circle is piled up along the height. So the
ratio [perimeter : diameter]=PI appears at the pyramid. |
3
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If you lay a vertical section through the centre of a pyramid parallel
to one square side, you get a (yellow) triangle. This triangle is almost
the main triangle of a pentagon.
It is special that two diagonals of a pentagon divide each other in
the golden ratio. |
You can often see a sky with five-cornered
stars on a blue background in tombs.
4
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The red triangle has the ratio chain
h' : h : a/2 = 5 : 3.90 : 3.11. This is nearly 5 : 4 : 3 with
5² = 4² + 3². The numbers 3,4,5 are Pythagorean
numbers.
Supposed: The triangle is a Pythagorean triangle. |
After the annual flood of the Nile the fields
were measured by 3-4-5-strings with knots.
5
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The base of the pyramid is lined up in the four points of the compass.
If you extend the diagonals of the base square, these lines include
the Nile delta. |
The position of the Great Pyramid is remarkable,
especially as you can look far into the Nile delta, if the smog of Cairo
permits.
6
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The Great Pyramid is not isolated. It nearly forms a straight line
with Chefren's and Mykerinos' pyramids. The constellation Orion's three
belt stars form (formed 4500 years ago) a similar line. Besides there are
two more pyramids, which correspond to two corner stars of Orion. (Source:
Robert Bauval, quoted in a report of the German TV station ZDF. |
Summary:
It is certain, that the ancient Egyptians chose the measurements of
the pyramid in order to make it safe and nice.
Who knows? Perhaps mysterious laws are hidden in the pyramids.
On the other hand: Numbers are patient... as we say in Germany.
The Power of Pyramids top
In the American magazine "Scientific American"
from June, 1974 a Dr.Matrix reported about a power, which came from pyramids.
Mysterious events happened in small models of pyramids: Razor blades became
sharp again, meat rotted slower, and a person sitting inside a pyramid
got supernatural abilities. These statements were proved by reports from
all over the world and seemed to be true.
Stop ;-) !
This was a scientific joke. The yearning for esotericism, which arose
at that time, should be satirized. - The famous journalist of "Scientific
American" Martin Gardner wrote this article.
I am interested, whether people sat inside small pyramids before
June 1974 ;-).
Calculations top
A pyramid is usually given by the square side
a and the height h. Other pieces, triangle side s, lateral faces
M, or volume V can be calculated by them. There are the following formulas.
s²=h² + a²/2
M²=a^4 + 4a²h²
V=1/3*a²h
You can generalize: If two of the five pieces a,h,s,M,
and V are given, the remaining pieces can be calculated.
There are ten cases:
1) Given: a,h. Search: V,s,M.
Solution: V=1/3a²h, s=1/2*sqr(2a²+4h²),
M=a*sqr(a²+4h²).
2) Given: a,s. Search:
h,V,M.
Solution: h=1/2*sqr(4s²-2a²), V=1/6*a²*sqr(4s²-2a²),
M=a*sqr(4s²-a²).
3) Given: h,s.
Search: a,V,M.
Solution: a=sqr(2s²-2h²), V=2/3*h*(s²-h²),
M=2*sqr(s^4-h^4).
4) Given: a,V. Search:
h,s,M.
Solution: h=3V/a², s=1/2*1/a²*sqr(2a^6+36V²),
M=1/a*sqr(a^6+36V²).
5) Given: h,V. Search: a,s,M.
Solution: a=1/h*sqr(3hV), s=1/2*1/h*sqr(4h^4+6hV),
M=1/h*sqr(9V²+12h³V).
6) Given: s,V. Search:
h,a,M
Solution: h³-s²h+3/2*V=0 und a^6-2s²a^4+18V²=0
must be solved:-(, then M=a*sqr(a²+4h²).
7) Given: a,M. Search:
s,h,V.
Solution: s=1/2*1/a*sqr(M²+a^4),
h=1/2*1/a*sqr(M²-a^4), V=1/6*a*sqr(M²-a^4).
8) Given: h,M. Search:
s,a,V.
Solution: s=1/2*sqr[sqr(4M²-16h^4)],
a=sqr[sqr(M²-4h^4)-2h²)],
V=1/3*h*[sqr(M²-4h^4)-2h²].
9) Given: s,M. Search:
h,a,V.
Solution: h=1/2*sqr[sqr(16s^4-4M²)],
a=sqr[2s²-sqr(4s^4-M²)],
dann V=1/3*a²h.
10) Given: M,V. Search:
a,h,s.
Solution: a^6-M²a²+36V²=0
und 12Vh³-M²h²+9V²=0 must be solved:-(, then s=1/2*sqr(2a²+4h²).
(Thanks to 10b in 1992/93)
An Example: (Give M,V):
Which shape has a pyramid, which has the same
volume and the same lateral faces as the Great Pyramid of Gise?
Solution: You derive the equation h³ - (M²/12/V)*h²
+ (3/4)*V = 0.
You get with V=2646000 and M=87120 the
solutions h1=146.6, h2=171.4,
and h3= -79,0 (found with DERIVE).
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The height h2=171.4m is the second solution.
The square side a2=215.2m belongs to the height
h2. |
This isn't a new contribution to the pyramid research. This only is
fun.
Largest Pyramid top
You can headfirst lay a second pyramid (green) inside a pyramid. Its vertex
is in the centre of the base and its square is parallel to the base of
the large pyramid. If the green pyramid is very low (1) or pointed (2),
its volume is small. There is a pyramid between, which is maximum. It is
the pyramid on the right (5).
You find this pyramid in the following way.
You lay a yellow triangle (3) inside the given pyramid, and introduce
the square side x and the height y. The volume is V=1/3*x²y. With
the help of the equation h:(h-y) = a:x you get V(x)=1/3*hx²-1/3*h/a*x³.
You get x=2/3*a and further y=1/3*h by V'(x)=0.
Pyramids on the Internet top
German
Christian Tietze/Rico Hecht
Architekturmodelle
von Pyramiden
(Zur Ausstellung „Pyramide – Haus für die Ewigkeit“ ab 6. September
2001 im Römisch-Germanischen Museum in Köln)
Frank Dörnenburg
Rätselhafte
Pyramiden
Ingrid Huber (Hubsi's Lehrer Homepage)
Grundwissen
über Pyramiden
Wikipedia
Pyramide, Pyramide
(Bauwerk), Pyramide
(Geometrie)
English
Andrew Bayuk (Guardian's CyberJourney To Egypt)
The Great Pyramid
Eric W. Weisstein (MathWorld)
Pyramid, Square
Pyramid, Square
Pyramidal Number
FERCO
The Pyramids of
Guimar
Unter Leitung des Ethnologen Thor Heyerdahl entstand in Teneriffa
ein Pyramiden-Museum. Es wird die (umstrittene) Theorie belegt, dass es
einen Zusammenhang zwischen den Pyramiden in Ägypten und in
Mittelamerika gibt. Trotzdem: Ein interessantes und geschmackvoll eingerichtetes
Museum.
Kevin Matthews and Artifice, Inc. (greatbuildings.com)
Sources
on Great Pyramid of Khufu, Pyramide
du Louvre
Lee Krystek
Khufu's Great Pyramid
Tim Hunkler
The Great Pyramid as Proof
of God
Wikipedia
Pyramid, Pyramid
(geometry)
References top
(1) Lancelot Hogben: Die Entdeckung der Mathematik, Stuttgart 1963
(2) Martin Gardner: Die magischen Zahlen des Dr. Matrix, Frankfurt
am Main 1987
(3) Armando Curcio (Hrg.): Meilensteine der Archäologie, Herrsching
1987
(4) David Blatner: Pi, Magie einer Zahl, Reinbek bei Hamburg 1997
(5) FTI Touristik Publications: Reisebegleiter Ägypten, 2000?
(6) "Gibt es ein Geheimnis der Pyramiden?" Two TV reports from the
series "Querschnitt" by Hoimar von Ditfurth, ZDF (29.03.1976 und 05.04.1976,
repeated in 1991)
In these two TV reports Hoimar von Ditfurth reacted to Erich von
Däniken's bestseller "Erinnerungen an die Zukunft" with the speculation:
"The pyramids were built with the help of extraterrestrial beings".
The report said: "The ancient Egypts already had the abilities
to build the pyramids."
Comments top
I made this website after a sight-seeing tour through Epypt including
the pyramids of Gise in April 2001.
Gail from Oregon Coast, thank you for supporting me in my translation.
Feedback: Email address on my main page
This
page is also available in German
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
2001 Jürgen Köller
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