Rationalization of Combinatorial Design in Architecture for Microhousing

Home , Kisho Kurokawa, Tetromino

A thesis submitted to the

Graduate School

of the University of Cincinnati

in partial fulfillment of the

requirements for the degree of

Master of Architecture

in the School of Architecture and Interior Design

of the College of Design, Architecture, Art, and Planning

Paul Kim

B.S. of Architectural Studies in the University of Illinois, Urbana-Champaign

June 2011

Committee Chairs:

Christoph Klemmt, AA Dipl.

Udo Greinacher, M.Arch

Abstract

By taking a combinatorial design approach, this thesis explores spatial and programmatic arrangements of residential units to design microhousing for temporary residents living in urban settings. In order to explore spaces, it was critical to itemize a building’s spatial components. By distinguishing the essential and non-essential components of each space, necessary spaces are defined by the activities of daily living. Then I created boundaries around the essential components considering anthropometry and circulation suitable for microhousing. These essential action spaces became templates throughout the thesis.

A growth of architectural form is one of the most critical factors in this exploration. Spatial combinations from the smallest to largest scale have facilitated this study in generating an ideal microhousing community. Rules and constraints are implemented to keep layouts organized and forms intact. The initial goal to preserve spatial efficiency was carried throughout the exploration. As scales grew, the primary focus was shifted from maintaining spatial efficiency to preserving modularity of combinatorics.

The premise of this thesis was to analyze necessary spaces and to generate an architectural form by combining modular components. This thesis concerns with process of spatial combinations and analysis of their spatial efficiency followed by the analysis of generated forms.

The design proposal hopes to benefit public and private interests, including single residents, students, young professionals, rehabilitating patients, senior citizens, and visitors. This thesis can persuade future real-estate developers, architects, and contractors to engage in the combinatorial design of microhousing to advance its construction techniques, to meet the housing demand, to sustain housing affordability, and to encourage innovative design strategies. More importantly, this thesis desires to encourage the combinatorial design method to architectural students to explore and expand their knowledge in generating innovative system of modular combinations.

The thesis is an on-going process that encourages all participants to bring focus to a global scale. Wherever housing is needed, this method can greatly assist in supplying residences for everyone.

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Acknowledgements

I would like to thank my dad, my mom, and my sister for their never-ending support from miles away. I am only here because of you, and I hope to reunite with everyone once again.

I would like to thank my closest friends Byungcheon Chang and Jason Cheung for words and acts of encouragement. You have always been there for me, knowing I was not fully capable of doing the same for you then. Our altruistic friendship is incomparable to others’.

Lastly, I would like to thank Professors Christoph Klemmt, Terry Boling, and George Thomas Bible for sharing their knowledge and broadening my perspectives on architecture. Their expertise and patience in teaching deserve the utmost praise.

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Table of Contents

Abstract ------ii Acknowledgements ------vii List of Figures ------9 1. Introduction ------10 2. Related Works ------11 2.1 Combinatorics in Mathematics ------11 2.2 Combinatorics in Literature ------12 2.2.1 Alice’s Adventures in Wonderland ------12 2.2.2 Oulipo ------13 2.2.3 Rule and Constraint ------14 2.3 Combinatorics in Gaming ------17 2.3.1 Tetris ------17 2.3.2 StarCraft ------20 2.3.3 Block’hood ------21 2.4 Combinatorics in Architecture ------23 3. Case Study: Microhousing in San Francisco ------25 3.1.1 Metabolism ------25 3.1.2 Microhousing ------27 3.1.3 Modularity vs. Prefabrication ------27 3.1.4 Site: Mission Bay ------28 3.2 Design Approach ------29 3.2.1 Action Space ------29 3.2.2 Action Module ------32 3.2.3 Unit ------36 3.2.4 Unit Pair and Unit Cluster ------39 3.2.5 Cluster Level ------42 3.2.6 Cluster Wing ------44 3.2.7 Arrangement on the Site ------48 Evaluation ------51 Conclusion ------52 Glossary ------53 Bibliography ------55 Image References ------57

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List of Figures

Figure 1: Leonhard Euler...... 11 Figure 2: The Tea Party in Alice's Adventures in Wonderland...... 12 Figure 3: Oulipo (Ouvroir de literature potentiélle)...... 13 Figure 4: Tetrominoes...... 17 Figure 5: Tetromino compositions in 10x8 boxes...... 18 Figure 6: Right skew (D), left skew (S), and branch (B) Soma Cubes...... 19 Figure 7: StarCraft II characters...... 20 Figure 8: Block'hood by Jose Sanchez from Plethora-Project...... 21 Figure 9: (Top-left) Nakagin Capsule Tower fabricated Metabolism diagrams; (top-right) Nakagin Capsule Tower unit section diagram; (bottom-left) Nakagin Capsule Tower's circular non-operable window; and (bottom-right) Nakagin Capsule Tower's interior workspace...... 26 Figure 10: (Left) Buckminster Fuller's Dymaxion House (1945) and (right) Moshe Safdie's Habitat 67 (1967)...... 27 Figure 11: Site Analysis in Mission Bay, San Francisco, California...... 28 Figure 12: Action space component...... 29 Figure 13: (Left) a cooking space, (center) a prepping space, and (right) a dishwashing space...... 29 Figure 14: (Left) a dining space, (center) an eliminating space, and (right) a bathing space...... 30 Figure 15: (Left) a grooming space, (center) a hosting space, (right) a lounging space...... 31 Figure 16: (Left) a sleeping space, (center) a changing space, and (right) a working space...... 31 Figure 17: Sleeping module schemes...... 33 Figure 18: Bathing module schemes...... 34 Figure 19: Hosting module schemes...... 34 Figure 20: Action module schemes...... 35 Figure 21: Unit schemes...... 37 Figure 22: Unit schemes...... 38 Figure 23: Unit cluster conceptual diagram...... 39 Figure 24: Cluster schemes...... 40 Figure 25: Cluster schemes...... 41 Figure 26: Cluster level schemes...... 42 Figure 27: Cluster level schemes...... 43 Figure 28: Cluster wing composition...... 44 Figure 29: Cluster wing schemes ...... 45 Figure 30: A perspective of the "amoeba" composition. Units with single-loaded corridors...... 46 Figure 31: A perspective of the "ring" composition. Units with double-loaded corridors...... 46 Figure 32: A perspective of the "claw" composition. Units with double-loaded corridors and vertical circulations...... 47 Figure 33: Building style schemes...... 48 Figure 34: Building combination study...... 49 Figure 35: Site Plan. A combination of building modules...... 50

1. Introduction

“Combinatorics” or “combinatorial design” refers to creating new ideas and meanings by combining multiple variables.1 In this method, one can find quantitative and qualitative measurements by rearranging “modules.” This tool is useful in many professional fields, such as mathematics, literature, gaming, and architecture.

In mathematics, combinatorics deals with the study of finite or countable discrete structures.2 In literature, constraints are implemented when specific terms are rearranged or modified to be read with completely different messages for readers.3 In gaming, diverse modules with assigned functions are placed within a boundary to change its surroundings, which could lead a player to succeed or fail. In architecture, it is a tool for exploring spatial arrangements.4

With combinatorial design, the goal of this thesis is to design a community of microhousing with balanced density in Mission Bay, San Francisco. The intent is to meet housing demand by increasing the number of affordable housing units and to stabilize residential real-estate market in bay area, where housing cost is extremely high. More importantly, I plan to deliver a design preserving the core idea of combinatorics, a modular system, throughout various scales.

1. Enrique Sobejano “Combinatorial Architecture,” Springer-Verlag Berlin Heidelberg, 2011, https://link.springer.com/chapter/10.1007/978-3-642-23435-4_7?no-access=true. 2. Anders Björner and Richard P. Stanley, "A Combinatorial Miscellany," Department of Mathematics of the University of Massachusettes Institute of Technology, September 5, 2010, accessed March 5, 2017, http://www-math.mit.edu/~rstan/papers/comb.pdf. 3. Warren F. Motte Jr., Oulipo: A Primer of Potential Literature (Lincoln and London: University of Nebraska Press, 1986), 27. 4. Sobejano, “Combinatorial Architecture,” 56.

2. Related Works

2.1 Combinatorics in Mathematics

Combinatorics is a branch of mathematics “studying the enumeration, combination, and permutation of sets of elements and

the mathematical relations that characterize their properties.”5 Leonrd Euler, an influential Swiss mathematician of the 18th century, introduced much of modern mathematical terminology and notation, particularly for mathematical analysis within combinatorics.6 Although mainstream mathematicians view this area as “less respectable as other areas of the branch”, it quickly gained popularity with emergence of computer science.7 Since then, other branches of mathematics, such as algebra, geometry, probability, and topology, have found its value in their areas.

Figure 1: Leonhard Euler. There are many approaches and subfields of combinatorics in mathematics. To name a few, there are enumerative combinatorics, analytic combinatorics, partition theory, graph theory, Design theory, finite geometry, order theory, matroid theory, extremal combinatorics, probabilistic combinatorics, and algebraic combinatorics. Moreover, there are related fields, such as combinatorial optimization, coding theory, discrete and computational geometry, comibnatorics and dynamical systems, and combinatorics and physics.8 In many branches of mathematics and its application, one must know the number of ways of producing an outcome.

5. Eric Weisstein, “Combinatorics,” February 27,2017, accessed March 08, 2017, http://mathworld.wolfram.com/Combinatorics.html. 6. William Dunham, “Euler: The Master of Us All,” Washington, DC: The Mathematical Association of America, 1999. 7. Anders Björner and Richard P. Stanley, "A Combinatorial Miscellany," Department of Mathematics, September 5, 2010, accessed March 5, 2017, http://www-math.mit.edu/~rstan/papers/comb.pdf. 8. Miklós Bóna, A Walk Through Combinatorics (Gainesville, FL: University of Florida, 2011), 34.

2.2 Combinatorics in Literature

2.2.1 Alice’s Adventures in Wonderland

Alice’s Adventures in Wonderland, commonly referred to as Alice in Wonderland, written by English mathematician Charles Lutwidge Dodgson, under the pseudonym Lewis Carroll, tells a fictional story of a young girl’s encounter with various characters like Cheshire Cat, the Duchess’s baby, Caterpillar, and Mad Hatter.9 Combinatorics is present in this literature, where each character is a module with a specific role that affects Alice in a different way in her encounter. In this chapter “Advice from a caterpillar”, Alice has fallen down a rabbit hole and eaten a cake that has shrunk her. The Caterpillar, smoking a hookah pipe, shows Alice a “magic mushroom” that can restore her to her original size. He tells her that one side of the mushroom stretches her neck, while another shrinks her torso.

Therefore, she must eat exactly the right balance to regain her proper size and proportions. 10

Figure 2: The Tea Party in Alice's Adventures in Wonderland.

Every character has a unique personality that is absurd one after the next. Characters’ actions and words do not make sense alone, but as a whole, the story seems to align harmoniously, despite their chaotic personas.

9. Charles Lutwidge Dodgson, Alice's Adventures in Wonderland (London: Macmillan, 1865). 10. Melanie Bayley,” Alice's adventures in algebra: Wonderland solved,” New Scientist, December 19, 2009, https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/.

2.2.2 Oulipo

Oulipo is well known for their use of Combinatorics in French literatures. Founded by Raymon Queneau and François Le Lionnais in November 24, 1960, Oulipo, short for French Ouvroir de literature potentiélle – roughly translated as “workshop of potential literature” – was a casual gathering of writers and mathematicians, who sought to create literary works using different constraints. These constraints were techniques that triggered ideas and inspiration by playing with certain words to create different meanings. Since then, the Oulipo’s goal was “to discover new structures and to furnish for each structure a small number of examples.”11 An example of their work is explained in the next section.

Figure 3: Oulipo (Ouvroir de literature potentiélle).

11. Warren F. Motte Jr., Oulipo: A Primer of Potential Literature (Lincoln and London: University of Nebraska Press, 1986), 1-2.

2.2.3 Rule and Constraint

Constraint often has a negative connotation. People accept the rule; they tolerate technique; but they refuse constraint because it is viewed as an excessive, unnecessary rule.12 There is a hermetic boundary between two domains: the one wherein the observance of rules is a natural fact, and the one wherein the excess of rules is a shameful artifice. This arbitrary boundary must be challenged for better knowledge of the functional modes of language and writing. One must first accept to treat language as an object, free from its subservience to its obligation. By doing so, it will be clear that language is a complex system, in which various elements are at work, whose combinations produce words, sentences, paragraphs, or chapters. More importantly, it becomes apparent that constraint goes beyond rules to those who challenge and extract its potential out of its routine functioning, revealing its hidden resources. Assisted by constraint, the final goal must not be focused on its exhibition but rather an exploration of virtualities.13

Let us look at a sonnet by Mallarmé:

Le vierge, le vivace et le bel aujourd’hui Va-t-il nous déchirer avec un coup d’aile ivre Ce lac dur que hante sous le givre Le transparent glacier des vols qui n’ont pas fui!

Un cygne d’autrefois se souvient que c’est lui Magnifique, mais qui sans espoir se délivre Pour n’avoir pas chanté la région où vivre Quand du stérile hiver a resplendi l’ennui.

Tout son col secouera cette blanche agonie Par l’espace infligée à l’oiseau qui le nie, Mais non l’horreur du sol où le plumage est pris.

Fantôme qu’à ce lieu son pur éclat assigne, Il s’immobilise au songe froid de mépris Que vêt parmi l’exil inutile le Cygne.14

This is translated as:

Will lovely, lively, virginal today Shatter for us with a wing’s drunken bow This hard, forgotten lake haunted in snow By the sheer ice of flocks not flown away!

A swan that was remembers it is he Hopelessly yielding for all his fine show Because he did not sing which way to go When barren winter beamed its apathy.

12. Warren F. Motte Jr., Oulipo: A Primer of Potential Literature (Lincoln and London: University of Nebraska Press, 1986), 23. 13. Ibid, 58. 14. Ibid, 58-59.

His neck will shake off that white agony Space deals out to the bird that will deny, But not earth’s horror where the plumes are clamped.

A ghost whom to this place his lights assign, He stiffens in the cold dream of contempt Donned amid useless exile by the Swan. (WM)15

The writer then proceed to a haikuization of this sonnet, meaning only the rhyming sections will remain while the rest are erased. In a mathematical language, one will consider a restriction of this poem to its rhyming sections by adding subjective punctuation:

Aujourd’hui Ivre, le givre pas fui!

Lui se deliver… où vivre? L’ennui…

Agonie le nie, pris,

Assigne mépris le Cygne.

15. Warren F. Motte Jr., Oulipo: A Primer of Potential Literature (Lincoln and London: University of Nebraska Press, 1986), 187.

This is read as:

Today bow snow away!

He shows… to where? Bored…

Agony will deny clamped.

assign contempt the Swan.

By taking the ending rhyme words of each line, a reader can interpret an entire new meaning. The purpose of this exercise is to deliver a new message by deriving impactful terms from the original sonnet. The constraint neither strengthens nor undermines the original poem; it simply delivers independent message that allows anyone to contribute to interpretation.

2.3 Combinatorics in Gaming

2.3.1 Tetris

Combinatorics is present in many strategic and temporal games. One of the most familiar games that embraces combinatorics is Tetris. Two-dimensional tetrominos, made up of four squares, drop within a rectangular boundary. A player rotates and places random units and attempts to place units side by side, avoiding any void, in which only the perfectly fitted units in a horizon will be eliminated. Eliminated blocks allow more space for space for a player to continue playing; the longer a player continues, the faster the units drop, which makes it more challenging. Tetris is a battle against time and speed as with many strategic and temporal games.16

There are seven total units in Tetris, in which two have mirrored symmetry. Among the five free tetrominoes are: “straight polyomino” (I), four squares in a straight line; a square polyomino (O), a 2x2 square; “T-polyomino” (T), a row of three blocks with one added below the center; “L-polyomino” (J or L), a row of three blocks with one added below the right or left side, respectively; and “skew polynomino” (S or Z), two stacked horizontal dominoes with the top one offset to the right or left, respectively.17

By using all the tetromino units at least once, we can generate rectangular boards of various dimensions, 8x5, 10x4, and 10x8. In a 10x8 board, two constraints are implemented: (1) no piece may touch another piece of the same color along a side, and (2) every piece must touch another piece of the same color at a corner.

Figure 4: Tetrominoes.

16. Alexey Pajitnov, “Tetris,” 1985-2017, http://tetris.com/about- tetris/?utm_source=tdc&utm_medium=ftr. 17. Eric W. Weisstein, "Polyomino," Wolfram Research, Inc., 2017, http://mathworld.wolfram.com/Polyomino.html.

Figure 5: Tetromino compositions in 10x8 boxes.

From the exploration, one can see the following properties: (1) mirrored or rotational symmetry; (2) chains of length 5 of touching pieces, rather than 2 and 3; (3) various rectangles and squares within; (4) groupings of similar shaped pieces; and lastly, (5) solutions where various pieces or groups can be interchanged. There are 35 solutions to the 8x10 rectangle.18

Even though Tetris is represented in two dimension for its simplicity and ease of understanding, tetromino units can be represented in three dimension. Three new units, also known as Soma Cubes, can be added to the original seven: right skew (D), a block placed on top left of a three-bend composition; left skew (S), a block placed on top right of a three bend composition; and branch (B), a block placed on the top center of a three-bend composition.19 These new additions are variations of skew polynomino that brings depth of combinatorial design.

18. Toby Gottfried, “Tetrominoes Puzzles,” www.gottfriedville.net, accessed March 8, 2017, http://www.gottfriedville.net/puzzles/tetra/. 19. Eric W. Weisstein, “Soma Cube,” Wolfram Research, Inc., 2017, accessed March 10, 2017, http://mathworld.wolfram.com/SomaCube.html.

Figure 6: Right skew (D), left skew (S), and branch (B) Soma Cubes.

With three new additions, right skew (D), left skew (S), and branch (B) units, to tetromino units, there are more variations in combinatorial design. Using these tetracubes, the pieces can fit in a 4x4x2 box as shown below:

layer 1 : layer 2 S T T T : S Z Z B S S T B : Z Z B B O O L D : L L L D O O D D : I I I I20

The concept of Tetris and combinatorial architecture share similar values. In Tetris, the goal is to play the game as long as one can as the pace gradually increases; likewise, the life of a building is designed to last as long as it can until it deteriorates. Tetromino units have different spatial functions based on its size and shape, similar to residential units that have different spatial components that define its independence and dependence. Both begins with the smallest form of units that grow into compositional models until they reach their preordained boundaries. Lastly, an absence or misplacement of a single unit can affect the overall design spatially, functionally, structurally, and visually. The practice of unit arrangement is visible in more complex games like StarCraft and Block’hood.

20. Eric Weisstein, “Tetromino,” Wikimedia Foundation, Inc., 2017, accessed March 8, 2017, https://en.wikipedia.org/wiki/Tetromino.

2.3.2 StarCraft

Blizzard Entertainment developed one of the biggest franchised computer games called StarCraft. The premise of this strategy game is to collect resources like mineral and gas, to build offensive and defensive structures that produce heavy artilleries and foot soldiers, to dominate enemies’ territories. Every unit has a designated role that contribute to the goal of the game.21 Each structure. There are four species – the adaptable and mobile Terrans, the ever-evolving insectoid Zerg, the powerfully enigmatic Protoss, and the “god-like” Xel’Naga, and they all have different style of characters.

As opposed to Tetris, StarCraft does not focus on combining forms to generate new functions. One of the game strategies is dispersing members all over the open territories so that a player has more control over the enemies. Proximity and quantity of members are important to consider; too little group of members is weak against an attack, whereas too many in one area is risky due to consolidation. Since each member has its unique skill, it is critical to understand which member to use against enemies. For instance, Marines, ground offensive units, are strong against smaller army of Zerglings but weak against heavy tanks. Therefore it is a more effective strategy to mix members of different functions for versatility.

The success of combinatorics of StarCraft depends on a player’s knowledge and skill level of the units and their functions. Strategic combination and dispersion of units at correcting timing is critical is overcoming battles. Efficient use of time and resources is the basis for any strategic and temporal game.

Figure 7: StarCraft II characters.

21. StarCraft II: Legacy of the Void, Blizzard Entertainment, Inc., 2017, http://us.battle.net/sc2/en/legacy- of-the-void/#units.

2.3.3 Block’hood

Block’hood is another example of strategy game dealing with combinatorics. Created by a Los Angeles-based architect, programmer, and indie game designer Jose Sanchez, Block’hood explores “spatial and temporal relations of organic architecture.”22 With an emphasis on ecology and sustainability, Block’hood allows a player to design a vertical city by redefining the rules, or constraints, that exist in reality.

Similar to Sims, the goal of this sandbox video game is to avoid decay by improving the quality of a virtual life. In this game, the units are “coupled and aggregated to generate larger assemblies, describing meaning, performance, and function at different scales of configuration.”23 By purchasing and placing predesigned units with social, economic, cultural, ecological, and environmental functions, i.e. a single-person apartment, a store, a tree, a windmill, or even a cattle farm, a player can alter the level of resources, such as electricity, water, labor, fresh air, money, organic waste, consumer value, greywater, and leisure.24 The game encourages a player to think about a design strategy that benefits the overall life of the virtual world.

Figure 8: Block'hood by Jose Sanchez from Plethora-Project.

22. Jose Sanchez, "Combinatorial design," Acadia, 2016, 53. 23. Ibid, 46. 24. Jose Sanchez, “Block'hood,” Plethora-Project, 2017, http://www.plethora-project.com/blockhood/.

Block’hood stands out from the previous work of Jose Sanchez in a way that spaces are created with emphasis on units’ functions and their proximity as opposed to just forms. Sanchez’s form-driven previous works “use a finite set of units or operations that can be arranged in different configurations.”25 He points that that “combinatorial design is not just the study of possible permutations of parameters, what Kostas Terzidis calls ‘permutation design,’ where any given variable of a design problem establishes a degree of freedom that can be catalogued and cross referenced to other variables, yielding the solution space of a given system.”26 Sanchez argues:

The open-endedness of the system implies, in contrast to Terzidis ideas, that there is no possible optimization, as the solution space of permutations grows with each unit added at an exponential rate, becoming computationally impossible to search for an optimum.27

However, by understanding the interior components of the units, it can lead to optimum level of combinatorial design. Interior components can represent mechanical, electrical, plumbing, communication, and pedestrian circulations, and these are the driving forces of unit compositions, which will eventually lead to efficient and effective architecture. The process might require extensive computational assistance, but the amount of options generated for analysis is vast and valuable.

25. Jose Sanchez, "Combinatorial design," Acadia, 2016, 46. 26. Ibid, 46-47. 27. Ibid, 47.

2.4 Combinatorics in Architecture

Combinatorial design in architecture must take account of all aspects that makes a building. One of the key factors to consider is spatial exploration, where rooms are created and analyzed based on each other’s proximity and orientation. Although residents are more used to lateral movement rather than vertical movement, this exploration must go beyond what is norm. Circulation is another critical factor that acts as spatial bridges. We must redefine what a successful circulation is, not only based on wheelchair accessibility, but also based on creation of natural order taken by placement of units.

This exploration will implement the computational representations, mechanism, and processes that explain spatial performance across the full range of human experience that requires simultaneous theorizing at three levels: overarching architectural design and control mechanisms, internal application of architectural sub-components/modules, and knowledge.28

Each of these three levels of theory has its own associated qualitative and quantitative parameters, and each of those internal spaces is modular. The execution of any module involves committing to a location at the intersection of these three very large (perhaps infinite) theoretical state spaces and consequently evaluating the extent to which that location in the joint state space is an appropriate scenario of human performance in that situation. If the scenario is deemed inadequate, then the search is on for a different combination of qualitative and quantitative parameters that will provide a better scenario.29

A thorough evaluation of even a modest portion of the total possible space will require an unprecedented amount of computing power because of the combinatorics associated with searching a multi-dimensional space consisting of (1) additional architectural components or modifications to existing components, (2) changes to numerical parameters, (3) changes to the initial knowledge state (including new task strategies) of the modules at the beginning of a run, (4) changes to the simulated task environment, or (5) all possible combinations of those things. Seemingly innocuous assumptions and implementation decisions can have dramatic consequences that interacts with a surrounding environment.30

Jose Sanchez, an architect and developer of Block’hood, has undergone a research on combinatorial design from an architectural background. The term combinatorial design encapsulates notions of both permutation and combinatorics and uses the studies of discrete finite sets of units and their possible arrangements by an algorithmic or intuitive process. According to Sanchez, combinatorial design is a design strategy that starts from the definition and individuation of parts, describing an open- ended series of relations with one another. These parts will be coupled and aggregated to generate larger assemblies, describing meaning, performance, and function at different scales of configuration.31

Gilles Retsin and Manuel Jiménez García also study combinatorial design in regards to architecture and large-scale 3D printing. Developed at the University of South California, “contour

28. Kevin Gluck, Matthias Scheutz, Glenn Harris, and Jeff Kershner, “Combinatorics Meets Processing Power: Large-Scale Computational Resources for BRIMS,” Proceedings of the Sixteenth Conference on Behavior Representtaion in Modeling and Simulation, Simulation Interoperability Standards Organization, and Simulation (Orlando, FL: Simulation Interoperability Standards Organization, 2007), 73-83.F 29. Ibid, 75. 30. Ibid, 83. 31. Jose Sanchez, "Combinatorial design," Acadia, 2016, 46.

23 crafting is a process where concrete is extruded from a nozzle that is mounted on a large-gantry-like structure.” Some models have entered commercial market, producing a number of full-scale architectural prototypes. Spatial plastic extrusion is a process where a robot arm extrudes plastic in the air, rather than in horizontal layers.32

32. Jose Sanchez, "Combinatorial design," Acadia, 2016, 47.

3. Case Study: Microhousing in San Francisco

3.1.1 Metabolism

Combinatorial design has its influence in Metabolism, a post-war Japanese architectural movement that combined two ideas of architecture and biomimicry with emphasis on organic biological growth. Derived from the Japanese word shinchintaisha, the literally translated term metabolism symbolizes the essential exchange of materials and energy between organisms and the exterior world. The group’s manifesto Metabolism: The Proposals for New Urbanism was published at the World Design Conference in 1960.33 The manifesto opened with the following statement:

Metabolism is the name of the group, in which each member proposes further designs of our coming world through his concrete designs and illustrations. We regard human society as a vital process - a continuous development from atom to nebula. The reason why we use such a biological word, metabolism, is that we believe design and technology should be a denotation of human society. We are not going to accept metabolism as a natural process, but try to encourage active metabolic development of our society through our proposals.34

Perhaps the most notable example of Metabolism was Nakagin Capsule Tower, constructed in Tokyo in 1972 by a Japanese architect Kisho Kurokawa. The capsules were designed with intent for mass production. A local shipping container factory prefabricated 140 capsules prior to their shipping to the site. Each capsule measures about 7’-5”(L) x 12’(D) x 6’-9”(H), providing about 90 SF of area and 621 CF of volume. These capsules can be connected and combined to create larger spaces to accommodate larger number of residents.35 The design concept allows flexibility and customizability based on a user’s individual preference.

Once a capsule is delivered to the site, it is lifted and connected to one of two core shafts by four high-tension bolts. The capsules are composed of welded lightweight steel-truss frames, cladded with galvanized ribbed steel panels, and coated with rust prevention paint and a glossy spray of Kenitex, an impervious weatherproof plastic with estimated 20-year life span.36 Tailored to remote salarymen, each capsule “contained a bed, storage cabinets, a bathroom, a color television set, a clock, a refrigerator, and an air conditioner.”37 Optional items like stereos were available upon request, allowing customization to resident’s functional preference.

Despite its functional innovation, Nakagin Capsule Tower had many failures. It was visually dense, giving cold, unwelcoming visual to incomers. Moreover, it was difficult to repair infrastructures like mechanical, electrical, and plumbing systems due to the nature of its modular design. In any case, if

33. Nakagin Capsule Tower: Japanese Metabolist Landmark on the Edge of Destruction, directed by Rima Yamazaki, performed by Nobuo Abe, Toyo Ito, Arata Isozaki, Toru Kawamura, and Kisho Kurokawa, Michael Blackwood Productions, Inc., 2015, documentary. 34. Zhongjie Lin, Kenzo Tange and the Metabolist movement (London: Routledge, 2010), 12. 35. Nakagin Capsule Tower: Japanese Metabolist. 36. Rem Koolhaas, Kayoko Ota, James Westcott, and Hans Ulrich Obrist, Project Japan: Metabolism Talks, (London: Taschen American, LLC., 2011), 34. 37. Hiroshi Watanabe, The architecture of Tokyo, (Stuttgart, London: Edition Axel Menges, 2001), 148-149.

25 an air conditioning was broken, the interior heat could not escape because the circular windows were not operable.38

38. Stan Alcorn, “These Photos Of Tiny, Futuristic Japanese Apartments Show How Micro Micro- Apartmnts Can Be,” Fast Company & Inc., September 25, 2013, https://www.fastcoexist.com/3017659/these- photos-of-tiny-futuristic-japanese-apartments-show-how-micro-micro-apartments-can-be.

3.1.2 Microhousing

Microhousing is a compact residential space that has a floor area of under 300 SF. It has become a necessary living option for many young professionals in metropolitan areas like New York, Seattle, and San Francisco, where living cost is extremely high. Despite its minimal size of residential area, microhousing offers affordability in rent, values simplicity in lifestyle, encourages engagement with neighbors, reduces energy consumption, and minimizes costs of construction.39 3.1.3 Modularity vs. Prefabrication

Modular construction and prefabricated construction are different but often overlap in parts – there are usually prefabricated components of a modular scheme, for instance. Prefabricated construction refers to any part that make up a residential unit, i.e. a structural wall assembly or a raised- floor system; whereas, modular construction refers to self-enclosed dwelling spaces, often composed of frame or structural assembly, fabricated off-site and assembled on-site.40 Size and form of modular units vary, but the components that go into assembling a unit are commonly mass-produced, meaning customization of units is easily done in off-site facilities.

Figure 10: (Left) Buckminster Fuller's Dymaxion House (1945) and (right) Moshe Safdie's Habitat 67 (1967). The idea of modular housing has been around for many years since Buckminster Fuller’s Dymanxion House in 1920s, even though it was not actually built until 1945. The Dymaxion House was Fuller’s solution to the need for “a mass-produced, affordable, easily transportable, and environmentally efficient house.”41 Though mass modular housing has had a few successes such as Habitat 67, the majority of projects either never make it past the planning stage or are resounding failures. Yet the Lego-like flexibility of modular construction has always inspired progressive architects. With the creation of new forms of housing and non-nuclear living situations, combined with advanced technological, the

39. "Microhousing is big in Seattle, and across the U.S.," The Seattle Times, 2014, http://old.seattletimes.com/html/homesrealestate/2023000906_hrehomeworkmarch02xml.html. 40. Kate Wagner, "The Modularity is Here: A Modern History of Modular Mass Housing Schemes," 99% Invisible, December 15, 2016, http://99percentinvisible.org/article/modularity-modern-history-modular-mass- housing-schemes/. 41. Jay Baldwin, “Dymaxion House,” New Scientist, accessed March 18, 2017, https://www.bfi.org/about- fuller/big-ideas/dymaxion-world/dymaxion-house.

27 conditions are ripe for a golden age of affordable modular housing – as long as zoning boards, cities, and individuals keep an open mind.42 3.1.4 Site: Mission Bay

The U.S. demographics of today have changed radically from those of 50 years ago, when the majority of urban housing was constructed for nuclear Baby Boomer families predominantly located in suburbs. As the younger generation emerge into professional world and transition from suburb to urban settings, businesses are following the trend to accommodate young entrepreneurs, who desire shorter and more sustainable commute between their households and workplaces by means of cycle and public transportation.43 San Francisco is a prime example of a city facing emerging young professionals and businesses, which lead to shortage of housing supply, rising housing demand and cost.

The proposed site for microhousing is located at Mission Bay, San Francisco surrounded by local businesses near South Street to the North and 16th Street to the South, waterfront across Terry A Francois Boulevard to the East, and University of California, San Francisco (UCSF), medical campus across 3rd Street to the West. AT&T Park, home to the MLB team SF Giants, is within a short public transit commute to the North. This former industrial site has a great potential to become a mixed-use community that includes microhousing residences and more local businesses for medical students, young professionals, rehabilitating patients, and visitors.

Figure 11: Site Analysis in Mission Bay, San Francisco, California.

42. Kate Wagner, "The Modularity is Here: A Modern History of Modular Mass Housing Schemes," 99% Invisible, December 15, 2016, http://99percentinvisible.org/article/modularity-modern-history-modular-mass- housing-schemes/. 43. Gretchen Livingston, "Fewer than half of U.S. kids today live in a 'traditional' family," Pew Research Center, (2014), http://www.pewresearch.org/fact-tank/2014/12/22/less-than-half-of-u-s-kids-today-live-in-a- traditional-family/.

3.2 Design Approach

3.2.1 Action Space

Before placing rules and constraints, it is important to introduce specific nomenclatures involved in understanding each component of a building. It is necessary to dissect a building to a smaller component–for instance, furniture or fixtures–that makes up a building. Below is the list of interior components that define spatial functions:

Figure 12: Action space component.

An action space is bounding space where a resident performs activities of daily living. Based on a function of an action space, there is at least one key component, such as a fixed or movable furniture, that will define a space for its primary intended use, even though some spaces have multi-functional intents. Although there are multiple options of interior components for an action space, the most typical item is selected as a default. More importantly, an appropriate volume of clearance is added to the dimension of an interior component. Therefore, the overall volume is the measurement of an interior component, for its primary function, and the measurement of clearance.

Figure 13: (Left) a cooking space, (center) a prepping space, and (right) a dishwashing space. 29

A cooking space generally requires an electric or a gas range to cook meals, which is its primary function, throughout the day. A range has an exhaust hood placed directly above the range for fume and odor exhaustion.

A prepping space is one of the most versatile spaces where a resident can adjust counter length according to its surrounding components. Most typical neighboring action spaces are cooking space and dishwashing space, which facilitate movements to and from one another. Depth and height of a counter may vary, but typically, they are more or less the same as the neighboring components.

A dishwashing space is strictly for washing dishes and cooking utensils. A dishwasher is an optional, luxury item; therefore, the space is not depicted as a default action space, and it is not included in the exploration of combinatorial design.

An important factor to consider about cooking, prepping, and dishwashing spaces is their accessibility to and from each other. Their placement, alignment, and orientation must act in unity for fluid movement.

A dining space needs a table and at least one chair to accommodate for eating meals through the day. Both of these items are ideally portable. A typical dining table has a circular, square, or rectangular profile in various sizes.

An eliminating space is for getting rid of bodily excrements. Unlike a toilet, a bidet is an optional, luxury item so it is not included in the default programmatic design.

A bathing space is where a resident washes his or her body. Both bathtubs and shower stalls are available in various sizes6”(L) x 3’-6”(D) x 8’-0”(H) in which a corner has a 45 degree deduction; and a large pentagonal shower is 4’-0”(L) x 4’-0”(D) x 8’-

Figure 14: (Left) a dining space, (center) an eliminating space, and (right) a bathing space. 0”(H), in which a corner has a 45 degree deduction. Lastly, square showers will share the same overall dimensions as pentagonal showers mentioned above without corner deductions. This exploration will use the largest shower stall and bathtub. An estimated area and volume of a shower stall are 12.25 SF and 85.75 CF, respectively. An estimated area and volume of a bathing space are 13.75 SF and 24 CF, respectively.

A grooming space is where a resident utilizes a sink or a vanity to groom daily. Activities include brushing teeth, washing face and hands, grooming hair, and applying make-up.

Figure 15: (Left) a grooming space, (center) a hosting space, (right) a lounging space.

A hosting space has the most amount of furniture. Since its purpose is not only to provide space for leisure but also to host guests, the space requires many furniture. Primary objects are for seating, which a resident can choose from several types among sofas, love seats, and chairs. Secondary objects are side tables for placing small objects like cups or lamps on top

Although a lounging space is an optional space, meaning one can survive without it, it is the most personal leisure space a resident can have. This space typically has a long chair next to a window that provides abundance of daylight.

A sleeping space needs a bed for a resident to rest overnight. For this exploration, beds come in three different sizes

A changing space is a private space where a resident changes in and out of appropriate clothes. This space has a closet and/or dresser. The dimension of closets is flexible as opposed to dressers. Containing both a closet and a dresser, a changing room will have an area and volume of 36 SF and 256.5 CF, respectively.

Lastly, a working space is an independent space curated for isolated productivity. Typically, this space includes a chair and a desk.

Figure 16: (Left) a sleeping space, (center) a changing space,31 and (right) a working space.

3.2.2 Action Module

An action module is a modular combination of two of more action spaces. There are five major action modules: sleeping, bathing, cooking, dining, and hosting. The most dominant action space will derive its nomenclature from its primary function. For example, sleeping is a more dominant action than changing or working even though they coexist in a single action module; therefore, “sleeping module” is a correct way to address the collection of action spaces within it.

As mentioned above, one of the most critical action modules is a sleeping module, which consists of sleeping space, changing space, and working space. The placement and proximity of a bed, a closet, and a desk can create a space commonly referred to as a “bedroom.” Scheme 2 of twin/single bed with an area of 49.73 SF has proven to be most spatially efficient, but the volume of a closet had to be minimized in order to meet the efficient value. On the other hand, Scheme 1 of king bed with an area of 98.33 SF allows comfortable space for two residents with plenty of closet space. Any scheme of sleeping module between the two ranges can suit individual’s need.

A bathing module is another critical one to explore. It contains a shower stall, a toilet, a sink, and a storage in different options. Scheme 2 with an area of 38.20 SF is the most spatially efficient option. In contrast, scheme 4 has more than double the area, equaling 106.13 SF due to excessive circulation that surrounds the action spaces, as opposed to bring in a central location. The difference of areas between scheme 2 and scheme 3 is only 1.5 SF. Despite having a larger area, scheme 3 provides a successful central circulation without the discomfort of a shower stall corner.

Lastly, hosting module deserves an exploration for having the most versatile spaces, including hosting space and dining space. Scheme 5 with a two-person sofa is the most spatial efficient option with an area of 66.84 SF; whereas, scheme 2 with a two-person sofa has an area of 111.38 SF. Both polar options are acceptable; however, the difference between the two varies based on a resident’s preference of circulation area, of which scheme 2 has the most. In the end, the importance of this analysis is not only to achieve spatial efficiency but to provide multiple options for eclectic residents.

Figure 17: Sleeping module schemes.

Figure 18: Bathing module schemes. Figure 19: Hosting module schemes.

Figure 20: Action module schemes.

3.2.3 Unit

A unit is a composition of action modules. It can be independent by having all action modules, consist of sleeping, bathing, cooking, dining, and hosting, for a single resident. Since there are five action modules, I have strategically designated four action modules into two pairs, leaving the remaining action module for dependent units. The order of combination does not matter in this exploration.

The first grouping of a unit dealt with combining dining-hosting spaces and cooking-sleeping spaces. Dining and hosting spaces are complementary since they both are flexible and versatile. Also, they facilitate engagement of entertainment, such as watching television, listening to music, or conversing with guests. In contrast to previous the pair, cooking and sleeping spaces are non-versatile; they have specific functions that cannot be substituted with many other actions. A resident sleeps and cooks at scheduled time throughout the day. Therefore, cooking and sleeping spaces are an appropriate pair due to process of elimination.

The second grouping deals with cooking-dining space and hosting-sleeping space. Cooking and dining spaces are complementary since one action follows the other in a fluid motion. On the other hand, a combination of hosting and sleeping spaces is contradictory since the two cannot function simultaneously, in which one requires active engagement, and the other requires isolation from interaction, respectively. However, when a resident and guests gather, one can argue that hosting- sleeping space can be successful on an occasional casual night-in.

The third grouping may produce the most controversial units. A bathing and hosting cannot coexist without a partition to divide the two spaces visually and acoustically, which in this case, neither are isolated. This combination is only problematic when guests are present or if any party feels uncomfortable. Nevertheless, this can be resolved by vertically separating the two spaces so that guests are at least visually away from their peripheral view.

Lastly, the final group includes a combination of cooking-dining space and bathing-sleeping space. Bathing and sleeping spaces are complementary because typically a resident will start the day by going to a bathroom to eliminate, bathe, and groom. This space is followed by cooking and dining spaces for breakfasts. This fluid transition from sleeping space to bathing space to cooking space to dining space is by far the most appropriate combination a resident could have if lacking a hosting space is not an issue.

Drawbacks of having complex level changes in units are having larger areas, which can be inefficient, and having limited wheelchair accessibility. Vertical changes in action modules meant more stairs and ramps must accommodate circulation. These transitional areas provide gradual physical and psychological shifts, allowing residents to ease in and out of spaces. Although stairs are limiting for wheelchair accessibility, there are plenty ADA units that are fully accommodative for people in wheelchairs, who can experience similar changes in perception due to shifts in ceiling heights. Even though default action spaces have designated floor-to-ceiling height, transitional spaces do not follow the same rule. Transitional spaces are more flexible since its function is to facilitate movement not only physically but also visually and psychologically.

Figure 21: Unit schemes.

Figure 22: Unit schemes.

3.2.4 Unit Pair and Unit Cluster

A unit can also be dependent by limiting two or more different action spaces in a group. When an action module lacks a certain functional space, a neighboring action module supplements for it, typically referring to bathing or cooking action modules. One type of grouping is known as a unit pair, in which case, both dependent units must rely on each other’s desired action modules, meaning neighboring residents must share and be mindful of each other’s bathing or cooking space. As a larger grouping option, a unit cluster deals with three units: one independent unit between two dependent units. Dependent units, lacking different action modules, must utilize an independent unit’s action modules. Both unit pair and unit cluster are designed to encourage interaction while preserving individual privacy. Furthermore, both models reduce building infrastructures, including mechanical, electrical, and plumbing components, construction cost of material, and assembly time.

Figure 23: Unit cluster conceptual diagram.

When creating a unit pair and a unit cluster, it is possible to have floor level changes. Changes in unit levels force corridors to align accordingly, assisted by stairs and ramps. Elevators are necessary for vertical circulation for wheelchair accessibility. Tectonic changes of corridors provide pedestrians unique visual experience during their commute.

Figure 24: Cluster schemes.

Figure 25: Cluster schemes.

3.2.5 Cluster Level

A cluster level consists of groups of units that fill the entire level of a building either by reflection or by rotation about an axis. Various extrusions of units’ exterior surfaces create various sizes of voids that can be open or enclosed spaces within a central space. An enclosed central space can be a hallway or a corridor, whereas an open space can be an atrium.

Figure 26: Cluster level schemes.

Figure 27: Cluster level schemes.

3.2.6 Cluster Wing

A cluster wing consists of a group of units stacked vertically. There are a row of units and a row of corridors that are vertically stacked. A cluster wing can be easily attached to the perimeter of a building. Since a cluster wing is segmented by column of three units, an effective system can be attached to any massing. Addition and subtraction of individual units may be necessary for fitting.

Figure 28: Cluster wing composition.

Figure 29: Cluster wing schemes

Figure 30: A perspective of the "amoeba" composition. Units with single-loaded corridors.

Figure 31: A perspective of the "ring" composition. Units with double-loaded corridors.

Figure 32: A perspective of the "claw" composition. Units with double-loaded corridors and vertical circulations.

3.2.7 Arrangement on the Site

As in every scale throughout the thesis, the concept of combinatorics and modularity is maintained. The building-to-building relations had to be analyzed for consistency and effectiveness.

Figure 33: Building style schemes.

Within a boundary of 4x8 grid, I generated three types of styles for a building’s form. The first style has hollow core for open atrium, possibly open to outdoor. The second style is enclosed to enhance density. And lastly, the third style has a more fluid form. I chose the second option due to its density and efficient use of space for microhousing.

After choosing a style of building’s form, I implemented a rule that buildings can be connect either tip-to-tip or surface-to-surface. Building modules are products of either rotation or reflection. Both connections allow continuation of internal circulation. The intent for this connection was to combine building modules to create indoor and outdoor spaces to balance congested microhousing units along the perimeter.

Figure 34: Building combination study.

Figure 35: Site Plan. A combination of building modules.

Evaluation

Combinatorial design in architecture is an appropriate method for exploration of spatial arrangement. In order for a designer to fully understand space, one should begin analyzing with the most intimate component a human encounters, which is the surrounding components – furniture and fixtures. By dissecting a building into smaller spaces, it was much clearer analyze their relationship with imbedded functional components that identified their primary intents. As the scale gradually increased, the combined spatial forms became more complex. New rules and constraints had to be implemented with broader perspectives, which shifted the focus of spatial efficiency to massing of the design, while preserving the premise of combinatorics. The study was a reverse progression a practice of architecture and construction.

The success of the spatial design varies based on the subjective views of the residents. Students, young professionals, rehabilitating patients, and visitors have different preferences on their living arrangements. This microhousing offers multiple options that can tailor their needs. Every unit is unique in regards to its layout of spaces, type of furniture and fixtures, and material choices, to facilitate comfort within individual unit. While residents hold independents from microhousing units, they are encouraged to engage with neighbors, by sharing specific spaces like bathrooms and kitchens. Sharing of spaces not only encourages residents’ engagement but also reduces materials, cost, and time used in assembly and construction.

Modular system allows units to stack on top another. Residential units are stacked on the perimeter of a building. Surrounded by stacked units of residential and commercial functions, central atrium is gives a massive open space for everyone to enjoy, especially for residents who wants to be out of their confined units. Corridors are structurally attached to modular units, giving the same tectonic shifts as the units’. Vertical and lateral shifting of units characterizes the overall building’s character. Every turn of the floors offer unique interior view in all direction. Even visual sensation residents feel can be modularly eclectic due to their surroundings.

As for the structural system, modular units are built with moment steel frames that residential units have moment steel frames that can support stacked units above. Above residential units have lighter frames than The structural frames of the bottom retail units are stronger to support the residential units that have lighter structural frames. Generally, the size and weight of structure is lighter as it reaches the top.

This exploration of combinatorial architecture is about the process and analysis. Throughout the extensive process, I was able to explore many options of spatial combination. Both conventional and nonconventional plan and sectional shifts provided different perspectives on spatial characteristics. It also allowed me to think about alignments of structure, infrastructure, and circulation.

The unit’s prefabrication and modularity is another successful strategy using combinatorial design. Mass-production of building materials allow cheaper cost of materials and shorter assembly and construction time; therefore, it offers affordable cost for residential housing. This is a great benefit for single residents, students, recent graduates, and young professionals who seek affordable housing in San Francisco, where housing cost is expensive. Rehabilitating patients and visitors can also benefit living in microhousing in Mission Bay since it is adjacent to the UCSF medical campus and several hospitals.

The effect of spatial combination varies based on functions of variables. No two spaces are alike, and each space has its own characteristic based on its function. Customization is important for preserving individual identify for electric residents. It is also important to tailor necessary spaces for an individual resident with respect to his or her needs and preferences. Conclusion

The combinatorial design in architecture offers large quantity of options for modular units. It offers diversity and versatility in spatial combination for future tenants, allowing customization to fit individual’s needs. The design is simple for construction, reducing time and cost of materials and assembly. The concept attracts architectural students to think beyond standard level of design and construction. In professional practice, it allows designers and contractors to improve upon current conceptual design. It encourages developers and investors to partake in this practice to meet housing demand in the U.S. real-estate market. Lastly, this thesis asks future participants to examine post- construction to see if spaces are truly effective for residents.

Glossary

Action module – a combination of two of more action spaces

Action space – a space defined by a resident’s actions of daily tasks

Activities of daily living (ADL) – routine activities that people tend to do every day without needing assistance

Block’hood – a neighborhood-building simulator, designed by Jose Sanchez, an architect/indie game developer based in Los Angeles, California, that celebrates the diversity and experimentation of cities and the unique ecosystems within them1

Combination – a joining or merging of different parts or qualities in which the component elements are individually distinct2

Combinatorial design / Combinatorics – (1) (in design) a term coined to describe non-parametric design strategies that focus on the permutation, combination and patterning of discrete units;3 (2) (in mathematics) the branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints, such as those of graph theory4

Efficiency apartment – an apartment in which one room typically contains the kitchen, living, and sleeping quarters, with a separate bathroom5

Enumeration – the action of mentioning a number of things one by one6

Functionalism – the principle that an architect should design a building based on the purpose of that building7

Metabolism – a post-war Japanese architectural movement that fused ideas about architectural megastructures with those of organic biological growth8

New objectivity – a movement in German art that arose during the 1920s as a reaction against expressionism9

Oulipian constraints – limiting rules set in a literature that will alter the original mecning10

Oulipo – short for Ouvroir de littérature potentiélle; roughly translated: “workshop of potential literature” is a loose gathering of (mainly) French-speaking writers and mathematicians who seek to create works using constrained writing techniques11

Permutation – the action of changing the arrangement, especially the linear order, of a set of items12

Shinchintaisha – literally translated as metabolism in a biological sense; symbolizes the essential exchange of materials and energy between organisms and the exterior world

Structuralism – a movement in architecture and urban planning evolved around the middle of the 20th century in reaction to rationalism, which led to a lifeless expression of urban planning that ignored the identity of the inhabitants and urban forms13

Unit – a collection of Action module that forms an independent single-person residence

Unit cluster – a collection of three units, who share two action modules of the same function i.e. bathroom or kitchen

Unit pair – a pair of units, which one is independent and the other dependent, allowing both units to share a specific action module i.e. bathroom or kitchen

Bibliography

Baldwin, Jay. “Dymaxion House.” New Scientist. Accessed March 18, 2017. https://www.bfi.org/about- fuller/big-ideas/dymaxion-world/dymaxion-house.

Bayley, Melanie. "Alice's adventures in algebra: Wonderland solved." New Scientist. December 19, 2009. https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra- wonderland-solved/.

Björner, Anders and Richard P. Stanley. "A Combinatorial Miscellany." Department of Mathematics, September 5, 2010. Accessed March 5, 2017. http://www- math.mit.edu/~rstan/papers/comb.pdf.

Dodgson, Charles Lutwidge. Alice's Adventures in Wonderland. London: Macmillan, 1865.

Dunham, William. “Euler: The Master of Us All.” Washington, DC: The Mathematical Association of America, 1999.

Gluck, Kevin, Matthias Scheutz, Glenn Harris, and Jeff Kershner. “Combinatorics Meets Processing Power: Large-Scale Computational Resources for BRIMS.” Proceedings of the Sixteenth Conference on Behavior Representtaion in Modeling and Simulation. Simulation Interoperability Standards Organization, and Simulation. Orlando, FL: Simulation Interoperability Standards Organization, 2007, 73-83.

Koolhaas, Rem, Kayoko Ota, James Westcott, and Hans Ulrich Obrist. Project Japan: Metabolism Talks. London: Taschen American, LLC., 2011, 34.

Lin, Zhongjie. Kenzo Tange and the Metabolist movement. London: Routledge, 2010, 12.

Livingston, Gretchen. "Fewer than half of U.S. kids today live in a 'traditional' family." Pew Research Center. 2014. http://www.pewresearch.org/fact-tank/2014/12/22/less-than-half-of-u-s-kids- today-live-in-a-traditional-family/.

"Microhousing is big in Seattle, and across the U.S." The Seattle Times. 2014. http://old.seattletimes.com/html/homesrealestate/2023000906_hrehomeworkmarch02xml.ht ml.

Motte Jr., Warren F. Oulipo: A Primer of Potential Literature. Lincoln and London: University of Nebraska Press, 1986.

Nakagin Capsule Tower: Japanese Metabolist Landmark on the Edge of Destruction. Directed by Rima Yamazaki. Performed by Nobuo Abe, Toyo Ito, Arata Isozaki, Toru Kawamura, and Kisho Kurokawa. 2015. Michael Blackwood Productions, Inc. Documentary.Pajitnov, Alexey. "Tetris." 1985-2017. http://tetris.com/about-tetris/?utm_source=tdc&utm_medium=ftr.

Sanchez, Jose. 2016. "Combinatorial design." Acadia, 44-53.

— “Block'hood.” Plethora-Project. 2017. http://www.plethora-project.com/blockhood/.

Sobejano, Enrique. “Combinatorial Architecture.” Springer-Verlag Berlin Heidelberg, 2011. https://link.springer.com/chapter/10.1007/978-3-642-23435-4_7?no-access=true.

StarCraft II: Legacy of the Void. Blizzard Entertainment, Inc. 2017. http://us.battle.net/sc2/en/legacy-of- the-void/#units.

Weisstein, Eric. “Combinatorics.” February 27,2017. Accessed March 08, 2017. http://mathworld.wolfram.com/Combinatorics.html.

Wagner, Kate. "The Modularity is Here: A Modern History of Modular Mass Housing Schemes." 99% Invisible. December 15, 2016. http://99percentinvisible.org/article/modularity-modern-history- modular-mass-housing-schemes/.

Watanabe, Hiroshi. The architecture of Tokyo. Stuttgart, London: Edition Axel Menges, 2001.

Weisstein, Eric W. "Combinatorics." Wolfram Research, Inc. 2017. http://mathworld.wolfram.com/Combinatorics.html.

— 2017. Polyomino. Wolfram Research, Inc. Accessed March 10, 2017. http://mathworld.wolfram.com/Polyomino.html.

— 2017. Soma Cube. Wolfram Research, Inc. Accessed March 10, 2017. http://mathworld.wolfram.com/SomaCube.html.

Image References

Figure 1: Leonhard Euler. https://upload.wikimedia.orsg/wikipedia/co mmons/d/d7/Leonhard_Euler.jpg

Figure 2: The Tea Party in Alice's Adventures in Wonderland.

Figure 3: Oulipo (Ouvroir de literature potentiélle). https://www.pinterest.com/pin/3940718393958898/

Figure 4: Tetrominoes.

Figure 5: Tetromino compositions in 10x8 boxes. http://rtomas.web.cern.ch/rtomas/Tetrominoes/

Figure 6: Right skew (D), left skew (S), and branch (B) Soma Cubes.

Figure 7: StarCraft II characters. http://www.sc2blog.com/wp-content/uploads/2010/08/battle-net-starcraft2-portraits-chart- w.jpg

Figure 8: Block'hood by Jose Sanchez from Plethora-Project. http://www.plethora-project.com/blockhood/

Figure 9: (Top-left) Nakagin Capsule Tower fabricated Metabolism diagrams; (top-right) Nakagin Capsule Tower unit section diagram; (bottom-left) Nakagin Capsule Tower's circular non-operable window; and (bottom-right) Nakagin Capsule Tower's interior workspace. http://architectureyp.blogspot.com/2011/05/nagakin-capsule-tower.html http://harboproject.blogspot.com/2009/08/nakagin-capsule-tower-tokyo-1972.html http://www.designboom.com/architecture/kisho-kurokawa-nakagin-capsule-tower-building/ http://www.designboom.com/architecture/kisho-kurokawa-nakagin-capsule-tower-building/

Figure 10: (Left) Buckminster Fuller's Dymaxion House (1945) and (right) Moshe Safdie's Habitat 67 (1967). http://www.interactivearchitecture.org/wp-content/uploads/2015/03/buckminster-fuller- dymaxion-house.jpg http://static.guim.co.uk/sys- images/Guardian/Pix/pictures/2015/5/11/1431336099014/e6509bd8-2da6-4b8f-8860- 5e175ecb7592-1020x612.jpeg

Figure 11: Site Analysis in Mission Bay, San Francisco, California.

Figure 12: Action space component.

Figure 13: (Left) a cooking space, (center) a prepping space, and (right) a dishwashing space.

Figure 14: (Left) a dining space, (center) an eliminating space, and (right) a bathing space.

Figure 15: (Left) a grooming space, (center) a hosting space, (right) a lounging space.

Figure 16: (Left) a sleeping space, (center) a changing space, and (right) a working space.

Figure 17: Sleeping module schemes.

Figure 18: Bathing module schemes.

Figure 19: Hosting module schemes.

Figure 20: Action module schemes.

Figure 21: Unit schemes.

Figure 22: Unit schemes.

Figure 23: Unit cluster conceptual diagram.

Figure 24: Cluster schemes.

Figure 25: Cluster schemes.

Figure 26: Cluster level schemes.

Figure 27: Cluster level schemes.

Figure 28: Cluster wing composition.

Figure 29: Cluster wing schemes

Figure 31: A perspective of the "ring" composition. Units with double-loaded corridors.

Figure 32: A perspective of the "claw" composition. Units with double-loaded corridors and vertical circulations.

Figure 33: Building style schemes.

Figure 34: Building combination study.

Figure 35: Site Plan. A combination of building modules.