Many buildings throughout history have incorporated geometric shapes inspired by mathematics, such as polyhedrons. Examples discussed in the document include Egyptian pyramids based on regular pyramids, the Pentagon building shaped as a pentagonal prism, and the Nakagin Capsule Tower composed of exchangeable cubic modules. Architectural structures showcase the interconnection between mathematics and building design across cultures and epochs.

1. GEOMETRIC
FIGURES
CAPTURED IN
PHOTOGRAPHIES

2. Mathematics in the field 
Polyhedra in architecture
 For many thousands of years, humankind has been trying to
provide itself a shelter. Building a house meant establishing
shape and order. The dwellings of modern man have a
polyhedral form, usually a regular prism. In every historical
epoch, architectural and mathematical cultures interpenetrate
and support each other. Traveling around the world, going to
work, school, shopping, we can notice the symbiosis of
architecture and mathematics at every step. It is an
indispensable part of our culture. Many buildings have a
shape typical of polyhedrons.
 Architectural concepts are often inspired by geometric
forms. An example are the Egyptian pyramids - buildings in
the shape of a regular quadrangular pyramid, serving as
tombs or the base of the temple. Further examples will be
presented in short descriptions for the next photos.

3. A triangle
The Mesoamerican step-
pyramid
The Great Pyramid of Giza

4. The Pentagon
Pentagon - a pentagonal prism; US Department of
Defense headquarters; built in the early years of
World War II.

5. The Gate of Europe towers
Also known as KIO
Towers (Torres KIO),
they are an interesting
example of
architecture. Those
are two inclined
prisms with a height
of 114 m and 27
floors, inclined in
relation to the ground
by 15°, which gives
the impression that
they lean towards
each other.

6. The Colosseum
 The Colosseum is of elliptical shape; an amphitheater
in Rome built between 70–72 and 80 CE. by the
emperors of the Flavian dynasty.

7. La Grande Arche de la
Défense
 A 110-metre-high
cube, it is a
reference to four-
dimensional
geometry; a
monument in the
business district
of Paris designed
to celebrate
humanity and
humanitarian
ideas.

8. Nakagin Capsule Tower, Tokyo,
Japan
 It is a residential and
office building with
exchangeable
modules in the shape
of cubes that form
13 floors. All modules
have the same
dimensions - 2.3 m ×
3.8 m × 2.1 m. The
modules can be
moved and
combined to change
the size of the space.

9. The dome of the Cathedral of
Santa Maria del Fiore in Florence
 It has an octagonal structure. Construction began in 1296, and the dome
itself was built in the years 1420–1436. The inner shell (skeleton) is made
of iron, while the outer shell is made of specially baked brick and stone.

10. The National Aquatics Centre
 also known as the Water Cube is an aquatics center at
the Olympic Green in Beijing, China. The aquatic center
is really a rectangular box (cuboid) 178 meters square
and 31 meters high.

11. Spaceship Earth
 Spaceship Earth is a dark
ride attraction at
the Epcot theme park at
the Walt Disney
World in Bay Lake,
Florida, USA.
The geodesic sphere is
a hemispherical thin-shell
structure (lattice-shell)
based on a geodesic
polyhedron.

12. The Lotus Temple
 Lotus Temple, Delhi, India - its appearance is inspired by the lotus
flower. The main pillars form a regular hexagon. The lotus flower
consists of 27 "petals".

13. The Shwedagon Pagoda
 The
Shwedagon
Pagoda,
Yangon, Myanm
ar - a conical
brick structure
covered with
layers of gold. It
is 99 m high
and set on a hill
51 m above the
city.

14. The Temple of Heaven
 Temple of Heaven, China - it is a tall, round edifice covered three-
level roof finished with a gilded ball. The whole building 38 m high
and 30 m in diameter, it rests on 28 large columns cedar without
walls.

15. The Leaning Tower of Pisa
 The Leaning Tower of Pisa is
one of the most famous
buildings in the world, visited
annually by about 10 million
tourists. In fact, it is the
cylindrical bell tower of the
cathedral and belongs to the
complex of Romanesque
buildings in Campo dei
Miracoli. The tower with a
height of about 55 meters,
currently tilted from the vertical
by 5 meters, each year
increases its deviation by 1
millimeter. Contrary to the
architects' calculations, the
tower has not collapsed so far.

16. The Eiffel Tower (pyramid)
 The Eiffel Tower is
a huge structure
on the Champ de
Mars in Paris,
supported by four
massive supports
connected by
arches, narrowing
towards the top
with a slender
truss structure.

17. The football house in Malawi
 The house looks like a football and has a
shape of dodecahedron.

18. Torre de Belém, Lisbon,
Portugal
 This is one of Lisbon's
greatest monuments, also
commissioned by King
Manuel and also in the
Manueline style. The
characteristic features of
this style, distinguishing it
from classical Gothic,
include semi-circular
arches of windows, details
in the form of elements
characteristic of shipping
(parts of ships, navigation
devices) or associated with
the sea (shells), lack of
symmetry, and finally
motifs taken from
traditional New World
architecture.

19. The Santa Justa Lift, Lisbon,
Portugal
 Supported by four
columns, the
structure measuring
45 meters in height
still serves both the
inhabitants and
numerous tourists
visiting the city. It is
built on a rectangular
plan.

20. The Casa da Música
 The Casa da Música is a concert hall in Porto, Portugal.
The building is shaped as a nine-floor-high
asymmetrical polyhedron covered in plaques of white
cement, cut by large undulated or plane glass windows.

21. The Rotunda of Saint Nicolas
(cylinder)
 The Rotunda of Saint Nicolas in Cieszyn is
a Romanesque rotunda located in the town
of Cieszyn, Poland. The structure was built in the 11th or
the 12th century as a Western Christian castle chapel
and stronghold (gord) church. A semicircular apse
adjoins the nave on a circular plan from the east,
separated from it by three steps. In the western part of
the nave there is a reconstructed gallery supported by
three column arcades. The stairs to the gallery run
inside the rotunda wall. On the other side of the
matroneum, there is a Romanesque portal, now bricked
up, which used to lead to the palatium - the dwelling of
the castle commander. The rotunda, as the only brick
building, probably served as the most important point of
resistance in the event of war.

22. The Rotunda of Saint Nicolas

23. The Warsaw Spire
 The Warsaw Spire is a
complex
of neomodern office
buildings in Warsaw,
Poland. It consists of a
220-metre main tower
with
a hyperboloid glass
facade, Warsaw Spire
A, and two 55-metre
auxiliary buildings,
Warsaw Spire B and
C. The main tower is
the third tallest building
in Warsaw and also the
third tallest in Poland.

24.  Examples of the use of polyhedrons
in architecture could go on for a long
time. From any trip we can bring
unforgettable mathematical
impressions because Maths - the
„mother of all sciences" accompanies
us at every single step we make.

25. Thank you!