# fract

###### BACHELOR thesis

###### / STUDIO FLORIÁN / FLO(W) /

###### CZECH TECHNICAL UNIVERSITY PRAGUE

###### FACULTY OF ARCHITECTURE AND URBANISM 2012 - 2014

###### Family house FRACT project aims to respond to the monotony of geometric objects. It tries to come up with a new form. In the era of digital technologies, the project finds inspiration in the virtual nature. The shape is inspired by generating fractal functions of hypercomplex numbers. The family house with the office is the subject of this thesis. As a construction material, the composite sandwich was designed. The composite material allows us to construct a shape of the walls freely and maintain the generated shape and uniformity of the material. The house is located on a sloping estate above Karlovy Vary.

##### VIRTUAL NATURE

Our perception is caught in a three-dimensional Euclidean space. This space is for us the “reality.” We are in it. We orient inside it by using our senses and distinguish different stimuli. Everyone lives in his own reality, differ from any other’s. The digital world could be considered as one of the realities as well. A world formed with numbers 0 and 1, the world that surrounds us more and more and together – shapes our being. Today we manage our lives through information technology. We use the Internet. We have profiles on social networking sites, communicate via email and also earn money virtually. A man could live his life through the virtual world. Only a computer would be enough to do that. The digital world is able to perfectly imitate our “reality.” This is well captured in the Matrix movie. The digital space has unlimited potential. It is subjected only to the rules which we define ourselves. We are able to take advantage of its benefits and the result then transforms and use in our three-dimensional world. We have incredible power. How to create a digital model of complex shapes such as a structure of tree bark or smoke? Virtual Nature is not created by classical 3D modeling, where all shapes are modeled by hand, one by one, without the possibility of later changes of their properties. Formation of large complex structures is thus practically impossible or inefficient. To model in this way, the tree bark would be demanding. Fractal functions are used to mimic natural shapes. Prescription of these functions is simple, yet it is able to generate an unlimited amount of data of different values. The principle consists of iterations (repetitions) and recursion (defining an object using itself). Initial input data are inserted into a prepared function that generates a result. This result is then placed at the beginning of the string and the function calculates again. The process is repeated several times until we receive the required amount of similar data, which together constitute the result – a fractal. From a simple shape, we can easily obtain an intricate shape by modifying and propagating only one function.

**GEOMETRIC FRACTAL**

The effectiveness of using fractals is best shows on geometrical fractals. Geometric objects – as a line – are directly used as the input data. One of the most famous geometric fractals is Koch Curve – Snowflake. Fractal properties can be subsequently modified. If the conditions are added the function will generate a modified data. We can also experiment with solids. Platonic solids are suitable because of their regularity. Surfaces of the regular icosahedron can be variously rotated, duplicated, moved or expanded. With IT technology we are able to do several variations in a few moments. The digital world enables us to create without boundaries. We are not limited by the physical body mass, which complicates intersections of various surfaces.

**COMPLEX FRACTAL**

We can insert anything as input data – everything that we are able to enroll using a numeric code. In the seventies, Benoît Mandelbrot studied the properties of fractals consisting of complex numbers and its graphical display in the Gauss-plane. His research would be without the use of computers impossible. Generated series of binary numbers can be easily displayed in a plane. One of the most famous fractals is the Julia set. It is entered with the sequence in + 1=zn2 + c. The domain of definition is all the z numbers from the complex plane. We choose an arbitrary complex number c, which will characterize the set. Now, we find out each point divergence by continual raise and adding a constant c. If the number doesn’t diverge the point belongs to the set. Based on the number of iterations until the absolute value exceeds the defined value it is possible to assign a color to the point. The set is made up of an infinite number of points. It allows us to zoom in and thus fly through shapes “interminably.” All points are generated by the same functions and the results are recalculated by it again, the default patterns are similar to each. In the real world, there would be theoretically equivalent infinite zoom if we had unlimited observational technique. We can fly through space and zoom closer to the galaxies, systems, planets, surfaces and zoom more to the very atoms and on and on.

**HYPERCOMPLEX FRACTAL**

From 2D complex, fractals are easy to get into 3D. We just insert hypercomplex numbers into the formula instead of complex numbers. Hypercomplex numbers contain more imaginary parts. To generate three-dimensional fractals quaternions – four components numbers are used. Such number may be generally written as: q=r + ai + bj + ck, where r , a, b and c are real numbers and i, j, k are imaginary part. The imaginary part has a similar definition as in complex numbers, thus: i2=j2=k2=-1. There are, however, more mathematical relationship applied among them: ij=k , jk=i , ki=j; her=-k , kj=-i ;=- j. Individual components of the quaternion can be interpreted as coordinates of w, x , y, z-axes. To be able to display such a four-dimensional element in 3D space, we must subtract one dimension – merge two of them together. We intersect the fractal with several planes. At this level, the one component becomes dependent on the remaining three. To fully describe the 4D body using 3D solids need to do a whole series of these cuts. In each section plane of 4D objects, there are different 3D penetrations. Unfortunately, our world is understood as the 3D space and so it is difficult for us to imagine a multidimensional object. The result of the generation of hypercomplex fractals are the coordinates of individual points. Is it possible to perceive a cluster of points as the surface of the object? Of course, it depends only on the perception of detail. If you select an atom as a point, then everything in our world is made up of points. This can be applied to much larger objects. For the items, we choose the star. From an adequate distance, we were able to identify appearance as an object- the galaxy.