fora1
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2008-12-18 15:16
Case Study 2.4 Minimal Surface(极小曲面)System
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// 虽然感觉parametric urbanism还没有讲完,但是觉得这个暂时更有意思一些,parametric urbanism以后再补充吧
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// Minimal Surface也算是很经典的system了,运用也非常广泛,比如伊东丰雄的台中艺术中心就是Minimal Surface的一种。Minimal Surface System本身也可以分成很多种类,比如
Costa's minimal surface, Gyroid minimal surface, 悬链面, 螺旋面, Scherk 曲面, Enneper 曲面。
不同种类的Minimal Surface被广泛的运用在不同的地方,比如Costa's minimal surface和Gyroid minimal surface给建筑师提供了极其有趣的特殊空间逻辑和结构特性(伊东丰雄的台中艺术中心就属于这类)。
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// 首先Minimal Surface相关的基本定义:
平均曲率
“在微分几何中,一个曲面 S 的平均曲率(mean curvature)H,是一个“外在的” 弯曲测量标准,局部地描述了一个曲面嵌入周围空间(比如二维曲面嵌入三维欧 几里得空间)的曲率。
这个概念由索菲·热尔曼在她的著作《弹性理论》中最先引入。”
Minimal Surface(极小曲面)
一个极小曲面是所有点的平均曲率为零的曲面。
极小曲面的一个推广是考虑平均曲率为非零常数的曲面,球面和圆柱面就是这样 的例子。Heinz Hopf 的一个问题为是否存在曲率为非零常数的非球面闭曲面。球 面是惟一具有常平均曲率且没有边界或奇点的曲面;如果允许自交,则存在平均 曲率为非零常数的闭曲面,Wente 在1986年曾构造出这样的自交环面(陈维桓 2006, 4.6节)。
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// 下面是几种空间上很有趣的Minimal Surface System
Costa Surface
Costa-Hoffman-Meeks Surface
Lopez-Ros No-Go Theorem
Planar Enneper
Scherk with Handle
Karcher JD Saddle tower
Karcher JE Saddle Tower
Schwarz H Family Surfaces
Lidinoid
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// 关于Minimal Surface System的例子有很多,而且不同的Minimal Surface System有完全不同的应用。这次的例子集中在Costa's minimal surface System或者类Costa's minimal surface System中(因为我个人比较感兴趣这些System提供的空间)。 ////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
Example 2.4.1 Hybrid Space(Design by TME)
YME 是由三个人组成的Group,Yevgeniy Beylkin (USA), Margarita Valova (BUL), & Elif Erdine (TUR). (现在三人都在Zaha的事物所工作)下面的例子是2006年他们在AA DRL的毕业设计
他们的project在research阶段主要使用软件是Mathematica, 一款基于C语言的数学模型绘图软件
第一部分:Research
他们做了大量的关于Minimal Surface System的research(虽然Minimal Surface System不是他们发明的,但是我个人认为DRL这种做大量research的精神是非常值得肯定的,他们最终产出的结果往往只占他们表达的一小部分,而且最终精彩的部分往往是过程而不是结果)
Level surfaces are surfaces that represent the solution to scalar-valued functions of three independent variables.
The three independent variables can be thought of as the X, Y, and Z coordinates of a point in three-dimensional Euclidean space.1For YME, the most interesting level surfaces are the triply periodic minimal level surfaces (TPMLS) which are periodic in all three dimensions. The periodic surface divides the volume into two congruent non-intersecting subvolumes. Each of the subvolumes forms a continuous network inside the whole system. If there are two such continuous subvolumes the structure is called bicontinuous. In a more complex situation, when there are two triply periodic surfaces inside the system they divide the space into three congruent non-intersecting subvolumes and hence such structure is called tricontinuous.2 Structures containing more than two continuous subvolumes such as tricontinous, quadracontinous, etc., are generally classified as polycontinuous for their complexity.
The Triply Periodic Minimal Level Surfaces are minimal for they have zero mean-curvature at all points. While most minimal surfaces are self-intersecting, the TPMLS on another hand are without self intersections.3 This unique quality of spatial organization in the TPMLS forms the bases for YME’s research. In this section, most common families of the TPMLS, the D, G, iWP, N, P, and P_W are briefly explained and then mathematically mixed with each other to produce new surface families.
The blue and yellow colors illustrate the 2 interweaving, non-intersecting spaces in each module of the Triply Periodic Minimal Level surfaces.
YME investigates the architectural qualities of the Triply Periodic Minimal Level surfaces in order to transform this family of geometries from an architectural diagram into an occupiable space. In this respect, we have been optimizing the sections and surface areas of these surfaces in 3d modeling programs without losing their descriptive qualities.
In the following diagrams, surfaces were imported from Mathematica both with high and few amount of points, thereby resulting in smooth or mesh geometry. Each surface module, spanning between 0 to 2Pi in 3 directions, was cut in 0, Pi/2, Pi, 3Pi/2, and 2Pi coordinates.
The diagrams exploring the surface area studies focused upon the occupiable space that can be inhabited, by manipulat¬ing the surface areas in 3d modeling programs.
Sectional Studies showing spatial transformations for smooth surfaces. The yellow and blue colors illustrate the 2 interweaving, non-intersecting spaces in each module of the Triply Periodic Minimal Level surfaces.
In this section, most common families of the (TPMLS) are mathematically mixed with each other to produce new surface families. Level surface equations have the convenient property that they can be mixed with each other to produce new surface families. In the experiments depicted in this section, the Triply Periodic Minimal Level Surfaces are used as terms in the equations, creating surfaces that maintain properties of both terms. For a given pair of such terms, a 2-parameter family can be generated by parameterizing the equation with variables s and t as,
0 = s*term1 + (1 - s)*term2 + t;
so that s gives the relative weights of the two terms and t gives an offset.
This morphing technique is named as Mixing of Terms.
Hybridization is a further investigation on the manipulation of Triply Periodic Minimal Level Surfaces described in Morphing section. YME explores Triply Periodic Minimal Level Surfaces as a potential architectural diagram for spatial organization. In this regard, it is crucial to gain control over the parameters of these geometries with the aim of understanding their properties and capabilities. The Method of Mixing Terms explained in Morphing section has been a start in manipulating TPMLS by mixing the terms of various surfaces with the addition of a new parameter, s. In this section, the manipulation of the parameters in the functions of TPMLS and the functions obtained by the Method of Mixing Terms has been advanced by introducing new variables into the existing functions. The new variables can be named as the basic mathematical functions, namely the Square Root, Exponential, Natural Logarithm, Factorial, and Absolute Value. These functions have been inserted in various places in the existing TPMLS functions, thereby creating a range of geometries that individually have unique properties. By exploring how these new variables manipulate the surfaces when inserted in specific parts of the functions, YME has begun to understand the techniques of having full control over the TPMLS.
Progression towards Hybrid Species shows the process of creating spatial differentiation within the TPMLS. The main aim of the series of operations displayed here has been to break the regularity of the pure TPMLS and to create spatial richness reached through topological differentiation. For this purpose, YME has conducted a progression of operations with the TPMLS, namely Adding, Multiplying, Mixing of Terms, and Combinations, which have given way to the discovery of the Hybridization operation. Basically, Hybridization means the insertion of a function inside the variables of a TPMLS. This method, which does not exist in any of the previous operations, has been very significant in controlling the topology of the TPMLS. More explicitly, this control has been gained by the realization of what type of function should be inserted in which variable of the TPMLS formula.
In the following diagrams, the Gyroid (G) and Diamond (D) surfaces has been used as examples in explaining the operations that have been described above.
Diamond logx^2:1/y^2 specie analysis.
The names of the Hybrid Specie surfaces in the following 3 diagrams have been given according to which mathematical functions have been used to hybridize them.
General space organization based on tetrahedronal component structuring.
第二部分:Iteration 01
For YME, the characteristics of the occupied site have been the main trigger for organizing the spatial layout of Iteration 01. The site can be mainly defined as rooftops and a limited spot at the ground level that acts as part of a courtyard. The circulation core is the only element of the building that is connected to the ground serving as the main entrance. All the subsequent spaces located at the rooftops gradually morph horizontally from the vertical circulation core. The main method used for generating the surfaces is ‘Mixing of Terms’, where 2 TPMLS surfaces are used as terms in the following equation, 0 = s*term1 + (1 - s)*term2 + t. The equation produces surfaces that maintain properties of both TPMLS terms.
Stage 01. Step 01: Define Grid. The massing boundary of the proposal is built from a mathematically defined grid. The whole grid can be subdivided into cubes having 3.14 (Pi) meters dimension. All the surfaces continuously follow the subdivisions of the grid, pointing out to the cellular nature of the TPMLS surfaces. The grid is created according to the site constraints as well the spatial potentials of the TPMLS surfaces.
A closer perspective view showing the defined grid for Iteration 01.
Stage 01. Step 02: Generate Surfaces. The TPMLS surfaces are connected by the ‘Mixing of Terms’ method. Each surface fits in the accompanying subdivision of the mathematically defined grid.
A closer perspective view showing the TPMLS surfaces inside the grid.
Stage01. Step03: Deform the Grid & the Surfaces. The last step is to adjust the grid and thus the TPMLS surfaces inside it according to the physical boundaries of the site. In this way, Iteration 01 fills in perfectly the residual spaces between the existing constraints of its location.
A closer perspective view showing the adjusted grid and the TPMLS surfaces for Iteration 01.
We approach Wolfram Research and GymBox as 2 opposing worlds that collide, with 2 contrasting briefs and spatial needs. In this respect, the above diagram and the following diagram form the basis of organizing Wolfram Research and GymBox spaces. This diagram analyzes the types of TPMLS modules and their possible connections. There are basically 2 types of TPMLS modules: 1. Bicontinous spaces are formed by the Gyroid and Diamond surfaces that create 2 distinct spaces within 1 module. 2. Single spaces are formed by Neovius, iWP, and P surfaces that create 1 space within 1 module. These surfaces create interweaving spaces when several of them connect with each other.
This diagram hints at the site specific organization of the TPMLS surfaces in the vertical and horizontal directions. Surfaces with bicontinuous modules form the vertical core of Iteration 01, creating spaces where the 2 programs, Wolfram Research and GymBox, collide. This core then expands in the horizontal direction, morphing into surfaces with single modules. Surfaces with single modules act as spaces where Wolfram Research and GymBox can exist independently from each other. Thus, the vertical and horizontal organization allows both coexistence and independency for the programs.
第三部分:Iteration 02
Stage01: Distributing Space.
In order to achieve volumes with different heights inside the main building block in vertical direction, the grid is divided in sections as follows: 2Pi, 3Pi, 2Pi, 4Pi. Through out the whole of the base block one overall morphing operation is applied at the beginning of the manipulation process that holds all the surfaces in one sequence. The secondary treatment is more distinctive for selected regions.
This image shows the final outcome at the end of Distributing Space process.
Stage01. Step01: Define Grid. In Mathematica to plot a surface is nec¬essary to input limits for each direction – xmin, xmax, ymin, ymax, zmin, zmax. The first step YME did to distribute and orga¬nise the spatial system into the site is to define a site-specific grid giving the overall boundaries of the space to work within. The grid is extracted in 3ds Max from the edges of the existing buildings. Once the general volume is completed it is subdivided into modules of Pi so the grid can be transferred easily into Mathematica language.
A closer perspective view showing the defined grid for Iteration 02.
Stage01. Step02: Generate Surfaces. The second step starts with subdividing the grid volume according to use, function and general program – circulation core, entrance lobbies, working areas, etc. Following the spatial requirements according to their use and the grid limits, through a lot of experiments in Mathematica, the surfaces organising the spaces are produced and exported as .dxf files to 3ds Max.
Stage01. Step03: Deform the Grid & the Surfaces. The third and final step during the distribution process is deforming the grid and the mass body of the building to a minimum in 3ds Max to adjust to the site.
Stage02: Differentiating Space. There is a clear distinction between the spatial qualities of the two organizations. GymBox in yellow is more open with large spaces that enable different ways of exercising. Wolfram Research in blue has a mostly cellular environment where the emphasis is on small teams with specialized agendas. The three entrances face the main street – the blue entrance in the middle is for Wolfram Research, and the two yellow side entrances are for Gym Box. This time Hybridising is used as a method for TPMLS manipulation which is a more complicated process comparing to Iteration One.
This image shows the final outcome at the end of Differentiating Space process.
Stage02. Step01: Offset the Base Surfaces. The first step undertaken is to offset the base surfaces created in Distributing Space process in order to define the two individual spaces for Wolfram Research and GymBox.
Stage02. Step02: Differentiate the Surfaces. At the next step an additional morphing of the offsets is performed to introduce the specific spatial characteristics addressing their use – cellular spaces for Wolfram Research and open spaces for GymBox. As a consequence the two interweaving spaces start to intersect which wasn't intended and was considered at the beginning as a failure. Later it appeared to be a positive aspect giving the two spaces the opportunity to register one another creating visual connections.
Stage02. Step03: Generate Rooftops. The third step for this stage is the distribution of seperate, not-interweaved spaces for Gym Box and Wolfram Research on the rooftops of the alongside buildings.
Stage02. Step04: Deform the Grid and the Surfaces. The fourth and final step during the differentiating process is deforming the grid and the mass body of the building to a minimum in 3ds Max to adjust to the site.
Stage03. Step01: Separate Existing Facade. From the surfaces created during Differentiating Space process, the ones overlooking the main street are sliced. The width of these surfaces is 2Pi.
Stage03. Step02: Generate Facade Surfaces.
The facade surfaces are generated by morphing the sliced surfaces with a plane in order to achieve surfaces with different levels of closure. As can be seen in the image, the GymBox facade, shown in yellow, has more closure because exercising environments require more artificial lighting. On the contrary, Wolfram Research facade, shown in blue, has lots of openings because their working environment needs more sunlight in contrast to artificial lighting. In this way, an observer can feel that there are 2 separate yet interwoven environments in one building by looking at the facade from outside.
Stage03. Step03: Deform the Grid & the Facade Surfaces. The same deformation that was applied to the differentiated surfaces is applied to the facade surfaces in 3ds Max, so that they adjust to the site.
Iteration 02. Exterior Perspective.
The possible symbiotic existence of Wolfram Research and Gym Box could be enhanced by interpreting Hybrid Species not just as mathematical constructs, but more importantly as architectural entities that could then prove to have unique spatial characteristics to them. The specific processes of constructing Iteration 02, namely Distributing Space, Differentiating Space, and Generating Facade resulted in a much more developed and specific architectural means for spatial organization. While Iteration 1 achieved the general distribution of Wolfram Research and Gym Box spaces, it was still not possible to read the different spatial patterns occurring in the 2 distinct companies. On the other hand, Iteration 2 clearly shows the methods that can be applied to generate 2 completely different types of spaces. The challenge here has been to be able to formulate these methods mathematically while still keeping in mind the most necessary characteristics of architecture, such as enclosures, circulation, occupiable working and exercising areas, etc.
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// Example 2.4.2 台中歌剧院(Design by Toyo Ito)
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// Example 2.4.3 Maximilian's Schell (Design by Ball-Nogues Studio)
Models
fora1 edited on 2008-12-19 09:10
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