DRU Publication

"PROPORTION" HT40003 ARCHITECTURAL HUMANITIES STUDIO EF 170026780

YUAN MENG

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“Mathematics is the alphabet with which God has written the universe.”

---Galileo Galilei

Vitruvian Man (c. 1492) by Leonardo da Vinci

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CONTENT Introduction and Research Abstract Proportion in Architecture Theme Origin Geometry References Stage Assumption Module Idea

Geometry Relationship Exploration Modeling Process Function Imagination

Physical Model Development Physical Model Preparation Exploration on Geometry Proportion Imitation

Measure Machine Measure Machine Composition Measure Machine in Physical Measure Machine Exhibition

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1.Introduction and Research

Abstract

The famous philosopher and mathematician Russell once said: "Mathematics, if you look at it correctly, not only has truth, but also has supreme beauty, a kind of cold and strict beauty, which does not suit the weak aspects of our nature... It can be pure to the point of sublime..." The way to record Math and produce numbers is called 'measure'. Math is always related to architecture. The idea of 'a thinking machine' could also be 'a measure machine'.

One relationship between numbers is proportion, which calculates multiple relationship among numbers. In that Case, the machine could measure proportion among objects in different scales.

The principle of interpret proportion might by means of using golden ratio. The so-called golden ratio refers to a certain mathematical proportional relationship between various parts of an object, that is, if a whole is divided into two, the ratio of the larger part to the smaller part is equal to the ratio of the whole to the larger part, and the numerical ratio is about 1:0.618. If adds two numbers together, 1.618 is obtained. The golden ratio is recognized as the most aesthetically valuable ratio, and it is also the ratio that most resonates with aesthetics.

GOLDEN RATIO IN SQUARES

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1.Introduction and Research

Proportion in Architecture The golden ratio is said to have existed since the earliest ages of ancient Egypt, about five thousand years ago. Since ancient Greece, the golden ratio has been used as a basic model for the proportions of beauty.

The tombs of ancient Egyptian slave owners and the Pyramid of Giza were built by hundreds of thousands of slaves in thirty years. The largest one is about 146.6 meters high, with a square base, the side length of which is about 230.06 meters, built by principle of golden proportions. The huge limestone cone of the four tombs shows the supreme theocracy of the three generations of pharaohs, who are the slave owners.

UNITE D'HABITATION ELEVATION

The Unite d'habitation is about 165 meters, the elevation is divided into two sections horizontally by the outer wall of the stairwell, and the vertical section of the solid building above the overhead floor is divided into two sections up and down by the sky commercial street. After calculation, both the horizontal and vertical directions, the elevation is divided into two sections with the golden ratio. After determining the floor height and building height, Le Corbusier should use the golden ratio to determine the length of the building after synthesizing the building scale and site constraints.

PYRAMID OF GIZA AERIAL VIEW

The Parthenon in Athens, Greece, is famous for its well-proportioned design. The perfect elevation proportions, beautifully carve and delicate Doric columns are remarkable. The whole temple consists of forty-six marble columns with a diameter of 1.90 meters and a height of 10.40 meters. The width of the corridor is 2.70 meters from north to south and 3.30 meters from east to west.

EIFFEL TOWER WITH GOLDEN RATIO

The famous Eiffel Tower is also built according to the golden ratio. The tower covers an area of 10,000 square meters and is about 300 meters high. The antenna is 24 meters high, and the total height is 324 meters.

PARTHENON WITH GOLDEN RATIO

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1.Introduction and Research

Theme Origin

Geometry References

The initial idea about theme of the “thinking machine”, which is called as a “measure machine”, relating to proportion, comes from the film “Interstellar”. In the film, there are three kinds of picture compositions that attract me: symmetry, composition in thirds and the golden ratio. Among all of them, the golden ratio is the one picked out for research, which is related to math.

Since related to mathematics and calculation, the golden ratio could be connected to other geometries, such as the isosceles triangle, the equilateral triangle and the square. In other words, in order to form a “measure machine”, the system should be composed with many elements to measure more objects. Also, try to arrange a rule to combine the curves would form new geometry shape such as circle and rectangle.

In our daily life, the golden ratio exists in many aspects. In the comic creation, the author might draw the scenery through it, also in architecture construction and art painting. On creatures like cats and humans, we can even find the existence of it. In nature and the universe, it is even easier to find. The golden ratio always exists in our living environment, and has already permeated into our life.

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In order to develop the machine, more relationships need to be considered in 3D direction, the projection idea and the relationship between the geometry could be connected together to form a complete system. Try to pull the Fibonacci curve up would be the further thoughts. Since the curve is infinite, only part of it would be picked out to carry out solution.

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1.Introduction and Research

Stage Assumption

The stages of building a measure machine might be like these: First, build a system in 3D dimension, which would follow some kind of regulation to be further exploded. Then find relationships between objects in the real world with the system. The objects might be different in types and shapes. Finally, the machine will be constructed. Relationship between different objects and the machine would be observed through exhibition.

The sketches of the steps are drawn to match the property of this measure machine, it could be a machine for measuring painting, toy cars, a robot or even a butterfly.

SKETCH PROCESS OF MEASURE MACHINE

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2.Geometry Relationship Exploration

Modeling Process

To start with the model in 3D, the Fibonacci Curve is used as a basis to calculate related squares and circles. The circle series that are divided in different scales are colored in orange, and each of the circle is given thickness to be a ring. The rings are separated from each other to form distance and the ability to combine with other sequences. From the front view , the track of the Fibonacci could still be recognized.

CIRCLE SEQUENCE IN 3D

The squares around the curve are supposed to be entities, so all of them are pulled up to be cubes in different scales. It seems that the cube series form interesting height difference, so in the real world, if give functional consideration on them in the real world, they might be stairways, exhibition stages or even chairs for people to sit. The quantity of rings and cubes are both number eight. In order to build the Fibonacci Curve in 3D, the cubes that settle down on the ground are pulled up to form a gradual change in rotation. After generating two kinds of curves by the cubes, in blue and red respectively, if connect the 3D curves together, one surface will be created to match the geometry relationship between Fibonacci and squares, circles among different directions. The surface would work as a visual guide. The observer might explore it in different directions.

PROGRESS OF FORMING FIBONACCI CURVE IN 3D

Creating an interlude relationship by the cubes, rings and 3D surface elements forms the basis of the measure system. The sense of expanding and rotating on the system is produced.

To make the core stable and stand on the ground, another cube series and extra columns are used to support them. A basic model of the Fibonacci is propped up by the cube stages to show the audience the idea which is originated from proportion. Then, adding layers of the model, holes are opened on some sides of the cubes. Considering the possible materials of the device in real, the cube stages and the columns might be concrete. Steel might be applied to create the rings. As for the cubes with holes, they might be made in frosted glass. The 3D surface might be a plastic one.

METHOD OF HOLDING THE SYSTEM

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2.Geometry Relationship Exploration

Function Imagination

CUBE STAGE FUNCTION DIAGRAM

In order to figure out more usage of the cube series, the possible functions of the Fibonacci Stages are imagined in three ways.

The first function is a stage chair for people to sit and have their leisure times. People with different height and figure could find out the most suitable cube for them to use.

The second possible function is that, the stage will work as part of a stair. It might exist in somewhere that only has a small height difference. People could use it and go upstairs or downstairs to another place.

The last one might be a stage for exhibition, many objects would be settled on each block according to their scales. People need to walk around and observe clearly to find out all the exhibited objects, since some of them are hidden by bigger and higher cubes in several directions.

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3.Physical Model Development

Physical Model Preparation

TOOLS OF MODEL MAKING

SKETCH DATA OF PHYSICAL MODEL

Aiming for figuring out how does the idea of measure machine take place in the real world, the following stage is about exploring the elements by making massing models. The units and the scale of the physical ones need to be figured out. If choose the square ranking the third to be the unit, which means the length of it is “1”, then, according to the golden ratio, the length of the other squares are calculated. Considering the specific unit, centimeters would be a proper choice.

PRODUCTION OF CUBES

Series of sketches are drawn to decide on the data of each part and the tools to make models, also comes the idea of changing supporting ways of the core. For example, the columns could be taken place by the hanging structure, or the system would be supported by structures extending from the wall. Among the physical modeling, the hanging idea is put into effect.

To make the massing models, tools like ropes, wooden stickers, the iron wire and foam board in different thickness are chosen. The maximum amount of work is about cutting cubes and sticking each piece together. The white cubes are made at first. The series are continual, by the way, the biggest one and the smallest one form great contrast but well harmony in scale.

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RANDOM DISPLAY OF CUBES

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3.Physical Model Development

Exploration on Geometry Connecting the white foam cubes by following regulation of golden ratio, then gets the solid stages.

They are rough because of hand-made reason, but the proportion between each cube is delicate.

If pile up the solid cubes in gradual change, they will look like a small tower which has many layers.

The shadow of the white tower complete the shape of the entity as a bigger tower. The complementary relationship will transform into other types by moving the location of source of light.

The cube with wholes are made by PVC boards. Although the scale of them is the same as the former solid ones, they look like more light in the mass, and might create more possibilities since there are space inside of each cubes.

The ABS stickers are used to form the ring, at the junction place where the two sides of one sticker surround and meet with together, a short piece of iron wire is used to tie the junction tightly. Although roughly, the scale of ring is shown properly by this way. The shape of bigger ring is more accurate than smaller ones because of material nature.

Combining the cubes and the rings together, the core system of the measure machine is finished. The vacuum of cubes help observing more about rings.

The ropes and other iron wires with the wooden stickers are used to test the possibility of hanging the cubes. When holding the sticker, the smallest cube is even smaller than the nail. By contrast, the biggest one is even bigger than the hand of the holder.

MODELING PROCESS

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3.Physical Model Development

The following ideas to find the relationship is to create chaotic and disorder randomly to build up new systems, the rings could have intersections, the cubes shouldn't always align with each other.

The objects could be settled in slanting. The cubes might be piled up one by one or even been embedded into a bigger cube. The amount of object groups could be more than one, and might be four. Most of them give the observer a sense of sculpture.

MODEL COMBINING PROCESS

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3.Physical Model Development

Proportion Imitation

PROPORTION OF TOY CAT

Aiming for finding out whether the cube and ring groups are suitable for building up measure system or not, the in-depth exploration is implemented. Numbering the cubes in A to H according the law that the scales vary from the smallest to bigger, so does the ring. The proportion of the toy cat could be imitated by 2 Cube A, 2 Cube B, 4 Cube C, 2 Cube D , 2 Cube E and 2 Ring A. To build up the proportion system of the toy cat, the process is about making from the bottom to top, from left to right. Due to the symmetry property, the face of the toy might be taken place by two bigger cubes, while there are more layers and changes in terms of the ears.

MODELING PROCESS

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3.Physical Model Development

PROPORTION OF TOY TRAIN

The imitation model for the toy train is bigger than itself, however, the proportion of each parts is clear, and outline the rough sketch. The train is divided into three layers. There are 5 Cube C, 2 Cube D, 3 Cube E, 1 Ring A, 1 Ring C and 4 Ring D to build up the train system.

MODELING PROCESS

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3.Physical Model Development

The toy dinosaur is unique on the figure, and it is difficult to imitate the proportion by using cube series. In that case, the idea of extracting the frame comes out. Using the ring series to explore proportion of a toy dinosaur is more like to imitate its' skeleton and create sections in different part of one dinosaur.

The skeleton also produces beautiful shadow on the wall. The profile of a dinosaur is abstracted in this way.

MODELING PROCESS AND SCALE SENSE

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3.Physical Model Development

SCALE CONTRAST BETWEEN ALL THE TOYS AND CUBES

The house is more complicated on details such as the chimney, windows and doors. The shooting of all the toys being together give the sense of how big they are, comparing to the cube modules, and how different are they in construction styles respectively. The toy house model is built up with 5 Cube B, 5 Cube C, 5 Cube D, 5 Cube E, 1 Cube F, 3 Cube G, 2 Cube H and 1 Ring C.

MODELING PROCESS

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PROPORTION OF TOY HOUSE

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4.Measure Machine

Measure Machine Composition

7 sets of cube series and 2 sets of ring series are selected to develop the whole system of the measure machine, they are supposed to be hanging structures which could act with objects. From each direction, the Fibonacci Sequence could be recognized to make relationship with displayed objects. Each of the series surrounds together. Some of the series are set to be vertically displayed, others are displayed horizontally. The biggest cube and the smallest one would connect with the conterminous series. The specific combination of modules are different. Some are connected with rings. Some are hollow cubes, and objects are supposed to be inside. Some are solid cubes with another set on the top. The tool to hang them up might be fixed at the center of cubes.

3D MODEL EXHIBITION

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4.Measure Machine

Measure Machine in Physical

SIDE VIEW OF PHYSICAL MODEL

The biggest cube is built to hold the hanging system. The stage of constructing it is to finish columns and beam at each side and then connect the top board with beams together. The bottom of the column is not fixed strictly.

Therefore, the structure on the sides could rotate if give them stress. When they meet with each other in the middle part, the system is stable enough, since a triangle is formed viewing from the front. When putting the module series and the whole structure together, the sense of measure is stronger. Each of the cube series is different.

There is comparison between void and mass, vertical and horizontal. The work of hanging all cube series and circle series is complex. Most of them are hanged by ropes and iron wires. The ones that locate lower and inner are selected to be hanged first. Each of the series need to be marked and evaluated different balance points. As more series are hung under the board, it tends to be out of shape and form arch. So do the columns on two sides.

PROCESS OF MAKING PHYSICAL MODEL

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4.Measure Machine

Measure Machine Exhibition

OTHER VIEWS OF MEASURE MACHINE

After finishing the main body of the measure machine, the next step is to make connection between the machine and objects. The cube module could imitate the proportion of specific shape of toys. When combining all the modules together and putting things into it, the sense and scale of overall proportion belonging to toy house, dinosaur, cat or train comes out. Each of them is in different location to create different reaction.

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BIBLIOGRAPHY

IMAGES

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"PROPORTION"

YUAN MENG | STUDIO EF 42