Structural Morphology in Architecture - WordPress.com · Structural Morphology in Architecture SOLIDS ... (Perspectiva corporum regularium, book) ... %20platonicos.pdf

SMiA Structural Morphology in Architecture

Structural Morphology in Architecture SOLIDS

WHY ARE SOLIDS IMPORTANT?

They are the base to understand the

unconventional structural systems like:

• Tensegrity

• Deployable structures

• Reciprocal structures

• Tree like structures

The comprehension of solid’s geometry

allows to explore and apply the

structures with more efficiency.

Relation

between

form and

structure

TENSEGRITY ARCH - KENNETH SNELSON

STTUTGART AIRPORT

GERKAN, MARG & PARTNER GEODESIC EXPANDIBLE DOME- HOBERMAN

RECIPROCAL VAULT

TENSEGRITY POLYHEDRON


OUTLINE

1. BACKGROUND

2. BASIC CONCEPTS OF SOLIDS

3. CLASSIFICATION

4. REGULAR SOLIDS: PLATONIC SOLIDS

4.1 DUAL SOLIDS

5. IRREGULAR SOLIDS

5.1 ARCHIMEDEAN SOLIDS

5.2 CATALAN SOLIDS

5.3 PRISMS AND ANTIPRISMS

5.4 PYRAMIDS, DYPIRAMIDS AND DELTOHEDRA SOLIDS

5.5 JOHNSON SOLIDS

6. TOOLS


1. BACKGROUND

POLYHEDRONS AND GEOMETRY IN NATURE


1. BACKGROUND

Neolithic Polyhedra

2000 B.C.

Roman Polyhedra

500 B.C

POLYHEDRONS IN THE HUMAN CIVILIZATION

WE USE THEM TODAY


1. BACKGROUND

Renaissance

Luca Pacioli's portrait,

attributed to Jacopo de'Barbari

1495

Leonardo da Vinci

1452-1519

(draws for the divina

proportione, book)

Giovanni da Verona

1520

POLYHEDRONS AT THE ARTS


1. BACKGROUND

Alberto Durero

1471-1528

Wentzel Jaminitzer

1508-1585

(Perspectiva

corporum

regularium, book)



1. BACKGROUND

Johannes Kepler

(mathematician

astronomer)

1571-1630

(Cosmological

Theory)


Maurits Cornelis Escher

1898-1972

Polyhedrons between

1948 and 1954


1. BACKGROUND GEOMETRY IN ARCHITECTURE

EGYPT PYRAMIDS

2500 B.C.

LOUVRE PYRAMID, PARIS

1983-1989

Arch. I.M.Pei


1. BACKGROUND

GEODE, PARIS

1985

Arch. Adrien Fainsilber & Eng.

Gérard Chamaillou


1. BACKGROUND

PHILLIPS PAVILION, BRUSSELS

Mathematical Sculpture Sculpture


1. BACKGROUND

DENVER INTERNATIONAL AIRPORT

HAJ TERMINAL JEDDAH AIRPORT

SAUDI ARABIA

1981

Eng. Horst Berger

Eng. Fazlur Rahman, Skidmore,

Ownings& Merrill


1. BACKGROUND


1. BACKGROUND

EDEN PROJECT, UK

2001

Tom Smith & Arch. Nicolas

Grimshow


1. BACKGROUND

CLOUDY CITY, NEW YORK

2012

Arch. Tomás Saraceno


1. BACKGROUND

GIANT PEANUT STRUCTURE, ICD

University of Sttutgart

2014

TRANSPORTABLE HOUSING,

Colombia

2014

Origamitek





DEFINITION

A solid is a closed shape or form in

three dimensions.

Solids are composed by:

FACES

- Curved: ROUNDED BODY

- Flat: POLYHEDRON

EDGES

Intersection of two faces

VERTEX

Intersection of three or more faces

FACES

EDGES

VERTEX

***We will focus at the POLYHEDRONS***



CONCAVE

Are the ones that can't be supported on

one of their faces.

The plane surface of their faces cut the

solid in two.

CONVEX

Are the ones that can be supported on

all their faces.

The plane surface of their faces contain

the solid.

POLYHEDRONS



CONCAVE

Are the ones that can't be supported on

one of their faces.

The plane surface of their faces cut the

solid in two.

CONVEX

Are the ones that can be supported on

all their faces.

The plane surface of their faces contain

the solid.

POLYHEDRONS

***We will focus at the CONVEX POLYHEDRONS***


3. CLASSIFICATION


3. CLASSIFICATION

REGULAR IRREGULAR

PLATONIC SOLIDS

ARCHIMEDEAN SOLIDS

CATALAN SOLIDS

PRISMS AND ANTIPRISMS

PYRAMIDS, DYPIRAMIDS AND

DELTOHEDRA SOLIDS

JOHNSON SOLIDS

CONVEX POLYHEDRONS can be classified in two groups:

***All the CONVEX POLYHEDRONS fulfill EULER‘S formula***


3. CLASSIFICATION

EULER’S FORMULA

F + V = E + 2

F = number of faces

E = number of edges

V = number of vertex

EXAMPLES

4 + 4 = 6 + 2

8 = 8

20 + 12 = 30 + 2

32 = 32

6 + 8 = 12 + 2

14 = 14


4. REGULAR SOLIDS:

PLATONIC SOLIDS



PLATONIC SOLIDS was studied

mathematically by the Pythagorean

school in the ancient Greece (530 A.C.)

EMPEDOCLES associates four of the

solids with the natural elements.

PLATÓN associates the fifth element

with the universe, because of this, they

are named PLATONIC SOLIDS.

EUCLID formalizes them like

mathematical elements.

Are perfect because

all their faces, edges

and angles are equal.

EARTH

FIRE

UNIVERSE

WATER

AIR



NAMES OF THE POLYGONS According to number of sides

PENTA 5 Pentagon

HEXA 6 Hexagon

OCTA 8 Octagon

DECA 10 Decagon

DODECA 12 Dodecagon

ICOSA 20 Icosagon

NAMES OF THE SOLIDS According to number of faces

TETRA 4 Tetrahedron

HEXA 6 Hexahedron (Cube)

OCTA 8 Octahedron

DODECA 12 Dodecahedron

ICOSA 20 Icosahedron

CHARACTERISTICS:

• They are regular solids.

• Their faces are regular polygons

(angles and sides are equal).

• All their faces, edges and angles

are equal.

NAMES

The names of the polyhedrons are formed

by greek roots.

Number of faces + Hedron

(prefix) (root)

Example: A solid with four faces is named

Tetrahedron (Tetra=4, Hedron = face)



TETRAHEDRON (4 FACES)

Equilateral Triangles joined

by three in each vertex

HEXAHEDRON OR CUBE (6 FACES)

Squares joined by three in each

vertex

OCTAHEDRON (8 FACES)

Equilateral Triangles

joined by four in each

vertex

DODECAHEDRON (12 FACES)

Pentagons joined by three in

each vertex

ICOSAHEDRON (20 FACES)

Equilateral Triangles joined

by five in each vertex

GEOMETRIC STABILITY

3V=E+6 (SCHÄFLI)

ICOSAHEDRON

3x12=30+6

36=36




4.1 DUAL SOLIDS


4.1 DUAL SOLIDS

FACES VERTEX EDGES PLATONIC SOLIDS

TETRAHEDRON

HEXAHEDRON

OR CUBE

OCTAHEDRON

DODECAHEDRON

ICOSAHEDRON

4

4

6

6

8

12

8

6

12

12

20

30

20

12

30

The duality between

solids is the

correspondence

between the vertexes

of one particular solid

and the centre of the

faces of another solid.

Characteristics

between two dual

solids:

•They have the same

quantity of edges.

•The quantity of faces

of one solid

corresponds to the

quantity of vertexes of

the other.


4.1 DUAL SOLIDS

The tetrahedron is the

dual of himself.

To each face of the

octahedron

corresponds one

vertex of the cube

To each face of the

cube corresponds

one vertex of the

octahedron

To each face of the

icosahedron

corresponds one

vertex of the

dodecahedron

To each face of the

dodecahedron

corresponds one

vertex of the

icosahedron


4.1 DUAL SOLIDS


5. IRREGULAR SOLIDS





TRUNCATED

TETRAHEDRON

TRUNCATED

HEXAHEDRON

TRUNCATED

OCTAHEDRON

TRUNCATED

DODECAHEDRON

TRUNCATED

ICOSAHEDRON

CHARACTERISTICS:

• They are semi-regular convex

polyhedron.

• All their vertexes are regular.

• Their faces are different, but all of

them are regular polygons.

Archimedean Solids they was

studied by Archimedes, he found

and classified them. They are 13

totaly.

11 of this solids are obtained

cutting the vertexes of platonic

solids. This transformation means

that the solid is truncated.

Other two solids are obtained

separating and twisting the faces of

the Cube and of the Dodecahedron.



CUBOCTAHEDRON

Maximum truncation: Is formed by

the cutting of the vertexes in the

middle point of the polygon edge.

The maximum truncation of two

dual solids forms the same solid.

Example: By the cube and the

octahedron is obtained the

Cuboctahedron.



Truncated

Tetrahedron

Truncated

Hexahedron

Truncated

Octahedron

Truncated

Dodecahedron

Truncated

Icosahedron

Cuboctahedron

Icosidodecahedron

Small

Rombicuboctahedron

Small

Rhombicosidodecahedron

Great

Rhombicosidodecahedron

Great

Rombicuboctahedron Snub Cube

Snub

Dodecahedron


5.2 CATALAN SOLIDS


5.2 CATALAN SOLIDS

CHARACTERISTICS:

• They are semi-regular convex

polyhedron.

• Their vertexes are not regular.

• Their faces are uniform, but not

regular polygons.

The Catalan solids are named for

the Belgian mathematician, Eugène

Charles Catalan.

They are obtained adding a pyramid

on the faces of the platonic solid.

In total they are 13, and they are the

dual of the Archimedean solids.

TETRAKIS

HEXAHEDRON


5.2 CATALAN SOLIDS

TRIAKIS

TETRAHEDRON

TETRAKIS

HEXAHEDRON

TRIAKIS

OCTAHEDRON

PENTAKIS

DODECAHEDRON

TRIAKIS

ICOSAHEDRON

RHOMBIC

DODECAHEDRON

RHOMBIC

TRICONTAHEDRON

DELTOIDAL

ICOSITETRAHEDRON

DELTOIDAL

HEXECONTAHEDRON

DISDYAKIS

TRICONTAHEDRON

DISDYAKIS

DODECAHEDRON PENTAGONAL

ICOSITETRAHEDRON

PENTAGONAL

HEXECONTAHEDRON





PRISMS: They are obtained by the

connection of two polygons

(bases), this connection is made

with parallel faces.

ANTIPRISMS: They are obtained by

the connection of two polygons

(bases), this connection is made

with triangles. Also, they can be

obtained twisting one of the

polygons (bases) in relation to the

other.



PENTAGONAL

PRISM

HEXAGONAL

PRISM

OCTAGONAL

PRISM

DECAGONAL

PRISM

TRIANGULAR

PRISM SQUARE

ANTIPRISM

PENTAGONAL

ANTIPRISM

HEXAGONAL

ANTIPRISM

OCTAGONAL

ANTIPRISM

DECAGONAL

ANTIPRISM


5.4 PYRAMIDS, DYPIRAMIDS AND

DELTOHEDRA SOLIDS


5.4 PYRAMIDS, DYPIRAMIDS AND DELTOHEDRA SOLIDS

PYRAMIDS: They are obtained by

the connection of the polygon

(base) with one point.

DYPIRAMIDS: They are two equal

pyramids joined.

DELTOHEDRA: They are like the

dypiramids but their faces are

deltoids.

SQUARE PYRAMID

HEXAGONAL DYPIRAMID HEXAGONAL DELHEDRA


5.5 JOHNSON SOLIDS


5.5 JOHNSON SOLIDS

CHARACTERISTICS:

• They are convex polyhedron.

• Their faces are regular polygons,

but not all are the same polygons.

The Johnson solids are named for

the American mathematician, Norman

Johnson. Totally are 92 types.


6. TOOLS HOW?

BY MODELS

BY SOFTWARE


6. TOOLS

BY MODELS


6. TOOLS

BY MODELS


6. TOOLS

BY MODELS


6. TOOLS

BY MODELS RECYCLABLE MATERIALS


6. TOOLS

BY MODELS


6. TOOLS

BY MODELS

GEOMAG (magnetic

toy from Switzerland)


6. TOOLS

BY MODELS

MODULMAX


6. TOOLS

BY MODELS

POLYGONS


6. TOOLS

BY SOFTWARE CABRI


6. TOOLS

BY SOFTWARE POLY

../../NATALIA/MORFOLOGIA/Morfología/Pedagoguery Software/Poly/poly.exe


CONCLUSIONS • Solids & Polygons are the base to understand

geometry of unconventional structural systems.

• We have an important tools for FORMFINDING:

- Physical models.

- Digital models by software.


LINKS http://en.wikipedia.org/wiki/Polyhedra

http://www.geomagvideo.com/

http://www.modulmax.com/

http://www.cabri.com/download-cabri-3d.html

http://www.peda.com/poly/

http://www.youtube.com/watch?v=rzO1kkxlK9I (paper)

https://www.uam.es/personal_pdi/ciencias/barcelo/historia/Los%20solidos

%20platonicos.pdf (paper)

http://en.wikipedia.org/wiki/Polyhedra

http://www.geomagvideo.com/

http://www.modulmax.com/













http://www.youtube.com/watch?v=rzO1kkxlK9I











https://www.uam.es/personal_pdi/ciencias/barcelo/historia/Los solidos platonicos.pdf
















THANKS!!!!

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Structural Morphology in Architecture - WordPress.com · Structural Morphology in Architecture SOLIDS ... (Perspectiva corporum regularium, book) ... %20platonicos.pdf