This document discusses fundamentals of mathematics in architecture and geometry. It covers topics like funicular polygons, forces diagrams, and examples of structural designs that utilize funicular polygons including bridges and stadiums. It provides instructions for students to determine coordinates and internal forces for given funicular polygons under different assumptions of horizontal and maximum forces. It also includes images and diagrams to illustrate funicular polygons and structural examples.

1. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FUNDAMENTALS OF MATHEMATICS I

2. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

3. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
PREVIOUSLY ON MATHS 1...

4. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FUNICULAR POLYGON. SHAPE and FORCES
EXAMPLE 1. POSITION and FORCES MAGNITUDES
EXAMPLE 2. POSITION and FORCES MAGNITUDES
EXAMPLE 3. POSITION and FORCES MAGNITUDES
EXAMPLE 4. POSITION and FORCES MAGNITUDES
EXAMPLE 5. POSITION and FORCES MAGNITUDES

5. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FIRST MINUTE QUESTION
Username: First part of your CEU email
For example:
if your email is a.garcia245@uspceu.es
Your username will be a.garcia245
Password: SAME AS USERNAME

6. FIRST MINUTE QUESTION
If the horizontal force was 300 kN instead of 500 kN......
The force in the cable AB would be smaller than 539 kN
A
B
C
D The position of joint B will not change
The force in the cable AB would be bigger than 539 kN
The force in the cable AB would be the same 539 kN

7. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Poly Corporation Headquarters. Beijing 2007. SOM

8. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Poly Corporation Headquarters. Beijing 2007. SOM

9. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

10. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: CNN

11. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: Gaudí Work

12. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: Gaudí Work. Blog Estructurando

13. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FUNICULAR POLYGON
The figure formed by a light string hung
between two points from which weights are
suspended at various points.
A force diagram for such a string, in which
the forces (weights and tensions) acting on
points of the string from which weights are
suspended are represented by a series of a
djacent triangles.
McGraw-
Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The Mc
Graw-Hill Companies, Inc.
Source: The Art of Spansh Bridge Designs. Princeton University

14. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: Quque Goberna

15. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: Quque Goberna

16. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
1.- Determine in a geometrical way the coordinates of the intermediate points B, C and D of the funicular polygon
formed by the represented forces. Assume the side joints are A with coordinates [0, 4 m] and E [12 m, 6 m]. Assume
also that the horizontal component of all the forces is 40 kN.
2.- Determine the internal forces in all the cable segments
Seafarers Bridge - Melbourne, Australia
2009 –[Nicholas Grimshaw]

17. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

18. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
THERE ARE INFINITEPOSIBLE FORCES DIAGRAMS and FUNICULAR DIAGRAMS. WE NEED SOME REQUIREMENTS
IN THIS EXAMPLE WE HAVE TWO REQUIREMENTS:
- HORIZONTAL FORCE 40 kN
- THE FUNICULAR POLYGON SHOULD START at A AND FINISH at E

19. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
FIRST GUESS: NO MATTER ITS VERTICAL POSITION

20. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
FIRST GUESS: NO MATTER ITS VERTICAL POSITION

21. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
FIRST GUESS: NO MATTER ITS VERTICAL POSITION

22. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

23. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

24. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

25. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

26. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

27. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

28. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Puente de Bacunayagua Cuba - 1959 - [Luis Sáenz Duplace]
1.- Determine in a geometrical way the coordinates of the intermediate points B, C and D of the funicular polygon
formed by the represented forces. Assume that the horizontal component of all the forces is 500 kN.
2.- Determine the internal forces in all the cable segments

29. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 500 kN

30. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 500 kN
so the pole is 500 kN away from the vertical forces
HORIZONTAL FORCE = 500 kN

31. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

32. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

33. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Pliezhausen footbridge
Reutlingen, Alemania - 2002 - [Holzbau Amann GmbH]

34. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
1.- Determine in a geometrical way the coordinates of the intermediate points B, C, D, E and F of the funicular polygon
formed by the represented forces. Assume that the horizontal component of all the forces is 200 kN. Determine also
the internal forces in all the cable segments for this hypothesis.
2.- How would the funicular polygon vary if the máximum force in the cable segmenst was 250 kN? Draw the funicular
polygonfor thies hypothesis and determine the internal forces in all the cable segments.

35. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
1.- Determine in a geometrical way the coordinates of the intermediate points B, C, D, E and F of the funicular polygon
formed by the represented forces. Assume that the horizontal component of all the forces is 200 kN. Determine also
the internal forces in all the cable segments for this hypothesis.
2.- How would the funicular polygon vary if the máximum force in the cable segmenst was 250 kN? Draw the funicular
polygonfor thies hypothesis and determine the internal forces in all the cable segments.

36. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 200 kN
HORIZONTAL FORCE = 200 kN
so the pole is 200 kN away from the vertical forces

37. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 200 kN
HORIZONTAL FORCE = 200 kN
so the pole is 200 kN away from the vertical forces
FUNICULAR POLYGON

38. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 200 kN
HORIZONTAL FORCE = 200 kN
so the pole is 200 kN away from the vertical forces
FUNICULAR POLYGON

39. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 200 kN
HOW DOES THIS DIAGRAM CHANGES IF THE
MAXIMUM FORCE WAS 250 kN?????
FUNICULAR POLYGON

40. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 200 kN
HOW DOES THIS DIAGRAM CHANGES IF THE
MAXIMUM FORCE WAS 250 kN?????
FUNICULAR POLYGON
????

41. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
MAXIMUM FORCE = 250 kN
HOW DOES THIS DIAGRAM CHANGES IF THE
MAXIMUM FORCE WAS 250 kN?????
FUNICULAR POLYGON

42. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Today’s words?

43. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
OLYMPIC STADIUM. MUNICH. 1968-1972. FREI OTTO

44. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
OLYMPIC STADIUM. MUNICH. 1968-1972. FREI OTTO