在二維鑲嵌藝術的範疇中，主要以兩種形式呈現對於平面密鋪的研究，其一為均勻幾何圖形的組合結果，例如:正方形、正六邊形；另一種則為幾何圖形的變形延伸，如:鳥、魚互補圖形、蜥蜴圖等。而在三維堆砌的領域裡則以多面體的形態分類與幾何架構為主要的研究方向，其中並不包含多面體本身的變形與其透過堆砌所可能生成的空間形態。本研究架構在鑲嵌圖形的基礎上，試圖尋找出存在於二維平面鑲嵌與三維多面體堆砌之間的連結，並經由參數化模型重新建構此種形態邏輯，進而使得鑲嵌藝術能以更高維度的空間形式應用於建築領域當中。

本研究分成三個階段，利用不同的工具及架構進行鑲嵌與形態的連結。第一階段為基本轉換，利用最常見的正六面體及以四邊形為架構的鑲嵌圖形，透過建模軟體的繪圖指令針對此圖形於三維空間立體中的連結進行繪製；並以二維的堆砌路徑將平面密鋪的特性應用至三維空間中。第二階段則是透過參數化軟體進行演算法的編程，將第一階段的研究成果加以分析，釐清研究者於上一階段的圖形繪製邏輯與建構規則，以各種路徑演算法為基礎進行更多形態上的演化。在本階段中，延續鑲嵌圖形對於多面體的規則，以新的平面圖形進行形態演算法的編寫，試圖在此操作下尋找出可被參數化的部分，藉由參數的調整嘗試各種形態組合的可能性，最後透過迴圈機制的編程，將單元依據路徑的演算進行形態堆砌。第三階段則是在理解基本的鑲嵌圖形與多面體的連結和演算法編程的操作後，以截角八面體與複數個鑲嵌圖形進一步地針對圖形生成曲面單元的形態作更多的嘗試。設計操作依舊延續前面的架構，然而與第二個階段的差異在於圖形的數量與選用的截角八面體之形態堆砌；由於單元生成演算法的編程僅適用於正六面體，因此路徑生成演算法將由兩個迴圈:一為堆砌路徑的生成，另一個則是曲面單元上的圖形連續性(空間腔體連續性)的判斷所組成。最後的階段期望透過不同種類的多面體驗證及拓展鑲嵌圖形應用於空間形態生成的可行性與多樣性，並以參數化設計的方式讓使用者匯入建築或是單元的形態模型，再利用演算法重現不同組合形態下的鑲嵌圖騰。

在本論文的操作中，主要透過參數化建模軟體的幫助，實踐鑲嵌圖形應用於多面體架構下形態的生成結果，並期待在建築空間上提出一種以聚合為構成的模式。以單元形態的組合變化為基礎，探討繁複細節體現在空間局部與整體時所產生的可能形態；而在建構方式上，由於數位設計輔助工具的運用，使得關於幾何的資訊能夠被重整而可視化的進行分析與組合，協助設計者在進行相關操作時能夠快速、直覺地進行大量的模擬及運算。在空間生成的本質能夠化簡為繁的當代，數位設計工具所帶來的影響是巨大的，進而提醒設計者能夠藉由理解並分析數據、幾何等更純粹的資訊，提出以往難以企及或是發展的嶄新設計觀念，為建築與空間設計帶來更豐富的新樣貌。

There are two main presentations of 2-dimensional tessellation, one is the combination of regular polygon, such as: square, hexagon; and another one is the deformation of these polygons, such as: birds and fishes in《Sky and Water》or lizards in《Lizards》by M. C. Escher. Meanwhile the tessellating topic in 3-dimensional mainly reflect on polyhedral honeycomb and geometry framework, but not included the deformation of polyhedron nor the spatial form that could be generated by honeycomb behavior. The purpose of this dissertation is to seek the form generating possibility by 2-dimensional tessellation pattern, when using 3-dimensional honeycomb as framework. Moreover, pushes the art of tessellation applied in architecture in a higher dimension through rebuild the morphological logic using parametric modeling tools.

This dissertation is divided into three parts, using different tools and framework to seek the connection between tessellation and form. The first phase using square and cube as framework to present some tessellation patterns and turn these patterns into 3-dimensional cubic structure to create tiles that could  be stacked, then orient these tiles onto a planar route to examined the relation between tessellation patterns and form. The second phase developing algorithm using parametric software to analysis and summarize the results for previous phase, proceeding more development based on various path-generating algorithm. This part continues on the logic of generating tiles based on patterns that as mentioned in the previous phase, in addition added more patterns to find the possible generations that could be parameterize in this method, then put these variables into a loop to generate pattern and form.Having discussed how to construct a pattern-based form generated by programming, the final part of this research addresses ways of using truncated octahedron as framework to generate a minimal surface tile based on multiple patterns. Using a similar logic to create tiles,the differences are the pattern quantities, bounding polyhedral, and tile generating method. Due to the tile algorithm in previous phase only served for cube, instead of truncated octahedron, thus the form generating algorithm is seperated into two main parts, the first part is to adjust the tile-generating method could be applied in this chosen polyhedral. The second part is to seperates the form-generation algorithm into two loop compositions: one is for the path; the other one is to process the pattern continuity status on the minimal surface tile through determined commands. The final phase is to examine the diversity of tessellation patterns which could be applied in different polyhedron to generate minimal surface tiles,and proceeding stacking behavior whenever a model be imported, in addition applying path-generating algorithm to reproduce an aggregation form while keep slightly differences between parametric adjustments.

In this research mainly discusses algorithmic aggregation could be applied as a mode into architectural spatial generating, furthermore algorithm provides variables regarding to details emerged in local and global form based on tiles combinitions. In terms of generating method aspect, to create or analysis new geometric data which provide designer performing intuitive simulations,the geometry information could be digitized and visualized in parametric software .In the contemporary era while parametric software makes enormous benefits for designer to propose concept which newer or unseen in the past through understanding and analysing information. These new design concepts of development bring a richer new perspective to architecture and spatial design.

目錄

第一章  緒論
1-1研究動機1
1-1-1  對數位設計的興趣1
1-1-2  碩一聚合研究的延續1
1-1-3  演算法影響空間排列組合1
1-1-4  結合參數化設計於聚合研究2
1-1-5  參數化設計與數位製造非歐幾何2
1-1-6  探索建築的新模式、語言、形態2
1-2研究目的3
1-2-1  數位時代的巴洛克精神3
1-2-2  研究不同文化的平面密鋪與嵌飾3
1-2-3  探討參數化紋理、模式與空間的可能性4
1-2-4  簡單規則經過疊代後的繁複呈現5
1-2-5  重覆單元的同質性與多元性5
1-2-6  接觸可能的數位設計與製造技術5
1-2-7  尋找可能具有文化形式的空間雕塑6
1-3相關領域6
1-4研究方法7
1-5研究成果7

第二章  相關文獻
2-1相關案例8
2-1-1  Grid and Tessellation Studies 8
2-1-2  Cubic Honeycomb Studies 9
2-1-3  Screen Walls at Boston Society of Architects 10
2-1-4  Polyomino 10
2-1-5  Aggregated Octahedra 11
2-1-6  HyperCells：The Self-Assembling Cells Of Future 12
2-2相關設計師12
2-2-1  Rinus Roelofs 12
2-2-2  Evan Douglis 13
2-2-3  Heather Roberge 14
2-2-4  Erwin Hauer 14
2-2-5  Jose Sanchez 15
2-2-6  Michael Hansmeyer 15
2-2-7  Moshe Safdie 16
2-2-8  Yona Friedman 17
2-3相關書目17
2-3-1《Structure in Nature is a Strategy for Design》by Peter Pearce 17
2-3-2《Autogenic Structures》by Evan Douglis 18
2-4相關論文19
2-4-1《參數化模具：二維圖案至三維形態設計與製造》  林澔瑢19
2-4-2《幾合單元組合的參數化設計與製造》  賴允誼19
2-4-3《鑲嵌法之數位設計與構築》  馮詠瑄19
2-5相關程式與插件20
2-5-1  Anemone 20
2-5-2  Rabbit 21
2-5-3  Kangaroo 21

第三章  規則建構
3-1 Top-down圖騰操作22
3-1.1設計發想22
3-1.2設計操作24
3-1.3小結31
3-2 Bottom-up圖騰操作32
3-2.1設計發想與邏輯建構32
3-2.2生成演算法34
3-3 小結55

第四章  極小曲面圖騰演算
4-1 極小曲面單元繪製56
4-1-1 Millipede生成極小曲面56
4-1-2 Mesh Machine曲面優化59
4-1-3 截角八面體極小曲面61
4-2 圖騰演算70
4-2-1 演算規則與判斷71
4-2-2 圖騰演算法編寫72
4-3 小結85

第五章  結論與建議
5-1 設計操作與回顧86
5-2研究心得與結論87
5-3後續研究建議88

參考文獻91


圖目錄

1-1研究動機
圖1-1.1 2014年秋季數位設計作品1

1-2研究目的
圖1-2.2 Morocco Pattern 4

2-1相關案例
圖2-1.1 Escher tessellation 8
圖2-1.2 Truchet tiles與其變化9
圖2-1.3 Screen Walls at Boston Society of Architects 10
圖2-1.4 Polyomino 11
圖2-1.5 Aggregated Octanedra 11

2-2相關設計師
圖2-1.6 HyperCells:The Self-Assembling Cells of Future 12
圖2-2.1 Rinus Roelofs 13
圖2-2.2 Evan Douglis 13
圖2-2.3 Heather Roberge 14
圖2-2.4 Erwin Hauer 15
圖2-2.5 Jose Sanchez 15
圖2-2.6 Michael Hansmeyer 16
圖2-2.7 Moshe Safdie Habitat 67 16

2-3相關書目
圖2-2.8 Yona Friedman 17
圖2-3.1 Structure in Nature is a Strategy for Design 18

2-4相關論文
圖2-3.2 Autogenetic Strucures 19

2-5相關程式與插件
圖2-5.1 Rhinoceros 20
圖2-5.2 Grasshopper 20
圖2-5.3 Anemone 20
圖2-5.4 Rabbit 21
圖2-5.5 Kangaroo 21

3-1 Top-down圖騰操作
圖3-1.1 Truchet tiles 2-Dimentional 22
圖3-1.2 Truchet tiles in 3-Dimentional 22
圖3-1.3 單元堆砌23
圖3-1.4 單元與互補單元23
圖3-1.5 單元與互補單元放置與變形24
圖3-1.6 單元在平面的放置與堆砌24
圖3-1.7 單元在空間中的放置與堆砌25
圖3-1.8 Totem Type A 26
圖3-1.9 Totem Type B 27
圖3-1.10 Totem Type C 28
圖3-1.11 Totem Type D 29
圖3-1.12 Totem Type E 29
圖3-1.13 Totem Type F 29
圖3-1.14 Totem Type G 29
圖3-1.15 Totem Series Model Photograph 31

3-2 Bottom-up圖騰操作
圖3-2.1 互補單元生成邏輯32
圖3-2.2 Loft指令與點位順序關係33
圖3-2.3 互補單元生產33
圖3-2.4 Type A 路徑生成34
圖3-2.5 互補單元生成算法35
圖3-2.6 路徑預設資料36
圖3-2.7 九宮格堆砌圖騰演算37
圖3-2.8 Totem Type A1 39
圖3-2.9 Totem Type A2 39
圖3-2.10 Totem Type A3 40
圖3-2.11 Totem Type A4 40
圖3-2.12 Hilbert Curve路徑生成41
圖3-2.13 Hilbert Curve疊代紀錄41
圖3-2.14 斷面選取42
圖3-2.15 參數化Hilbert Curve 43
圖3-2.16 Hilbert Curve圖騰演算43
圖3-2.17 Totem Type B1,B2 44
圖3-2.18 Totem Type B3 45
圖3-2.19 Perlin Noise演算法46
圖3-2.20 Perlin Noise演算法47
圖3-2.21 Perlin noise圖騰演算47
圖3-2.22 Totem Type C1,2,3,4 48
圖3-2.23 貪食蛇路徑生成49
圖3-2.24替換輸入單元50
圖3-2.25貪食蛇路徑疊代50
圖3-2.25貪食蛇圖騰演算51
圖3-2.26 Totem Type D1 53
圖3-2.27 Totem Type D2 53
圖3-2.28 Totem Type D3 54
圖3-2.29 Totem Type D4 54

4-1 極小曲面單元繪製
圖4-1.1 Millipede 極小曲面試算56
圖4-1.2 以Truchet Pattern生成極小曲面於正六面體56
圖4-1.3 Truchet Pattern於六面體上旋轉生成紀錄57
圖4-1.4 Truchet Pattern於六面體上生成結果57
圖4-1.5 Truchet Pattern於六面體上的不同選轉向度極小曲面定義58
圖4-1.6 Weaver bird Loop subdivision 59
圖4-1.7 Minimal Surface Grasshopper Definition 59
圖4-1.8 Millipede+Weaverbird and Kangaroo Mesh Machine 60
圖4-1.9 Minimal Surface in Kangaroo Mesh Machine 60
圖4-1.10 Hexagon Truchet Tessellation 61
圖4-1.11 Pixelized Surface with Box & Truncated Octahedron 61
圖4-1.12 Truncated Octahedron Minimal Surface by Truchet Pattern 62
圖4-1.13 Truncated Octahedron Minimal Surface Grasshopper Definition 63
圖4-1.14 Truncated Octahedron Minimal Surface with 2 hexagon Patterns 64
圖4-1.15 Truncated Octahedron Minimal Surface with 2 Hexagon Patterns Definition 66
圖4-1.16 4 Different Pattern Orienting Methods66
圖4-1.17 Find Duplicate Units 67
圖4-1.18 Chosen Pattern 0 orienting method unit 68
圖4-1.19 Chosen Pattern 1 Orienting Method Unit 68
圖4-1.20 Chosen Pattern 2 Orienting Method Unit 69
圖4-1.21 Chosen Pattern 3 Orienting Method Unit 69

4-2 圖騰演算
圖4-2.1 模型單位尺度與圖騰形態的關係 & 蚱蜢定義70
圖4-2.2 Pattern Correcting 71
圖4-2.3 Cull Closest Point Anemone 72
圖4-2.4 Cull Closest pt n = 0 , n = 100 72
圖4-2.5 Plane & Surface 資料結構73
圖4-2.6 當前座標平面與過往集合的座標平面之交集73
圖4-2.7 利用Path Mapper整理當前座標平面成為過往的座標平面集合74
圖4-2.8 根據Pattern的正確性決定下個使用的單元74
圖4-2.9 找出交集的座標平面集合，並提出分別為何種Pattern作為後續判斷的參考75
圖4-2.10 利用Pattern判定的正確與否決定第二層迴圈的狀態75
圖4-2.11 TotemA Generating Process and Data 76
圖4-2.12 Truncated Octahedron TotemA 77
圖4-2.13 Panton Chair Tessellation 79
圖4-2.14 鑲嵌分析 Tiles No.為每一代使用的單元編號；Quantity表示每種單元的使用數量；Location則為單元的位置資訊80
圖4-2.15 Vault Totem A 與鑲嵌分析81
圖4-2.16 Vault Totem B 與鑲嵌分析83

5-3後續研究建議
圖5-3.1 Isosurface Definition 88
圖5-3.2 Multiple-Polyhedron Honeycomb 89
圖5-3.3 3D-Printing Truncated Octahedron Honeycomb using Magnet 90

參考文獻

【相關書目】

Peter Pearce
1978《Structure in Nature is a Strategy for Design》,MIT Press。

Evan Douglis
2013《Autogenic Structures》,Taylor & Francis。

Syed Jan Abas、Amer Shaker Salman
2004《伊斯蘭的幾何藝術》廖純中譯,左岸文化出版社。

彭智謙、湯天維、陳珍誠
2015《蚱蜢狂熱 I》,淡江大學出版中心。

【相關論文及期刊】

林澔瑢
2013《參數化模具：二維圖案至三維形態設計與製造》,淡江大學建築學系碩士論文。

賴允誼
2013《幾合單元組合的參數化設計與製造》,淡江大學建築學系碩士論文。

馮詠瑄
2015《鑲嵌法之數位設計與構築》,淡江大學建築學系碩士論文。

李明翰
2015《應用形態發生學於形態找尋》,淡江大學建築學系碩士論文。

【參考電子文件及網站】

Amalgamma
2015《Fossilized 》,GAD Portfolio // Wonderlab :: Research Cluster 4:
https://issuu.com/amalgamma/docs/amalgamma_portfolio

The Bartlett School of Architecture UCL
2015《MArch Architectural Design (AD) 2015》:
https://issuu.com/bartlettarchucl/docs/bart_ad15_issuu

The Bartlett School of Architecture UCL
2016《MArch Architectural Design (AD) 2016》:
https://issuu.com/bartlettarchucl/docs/bpro_ad16_aw_issuu_1_

Dechen Zeng
2017《Polyomino 3》,Research conducted at USC :
https://issuu.com/josesanchez010/docs/dechenzeng_fa2015_402a_portfolio


Jingbo Yan
2017《Polyomino 3》,Research conducted at USC :
https://issuu.com/josesanchez010/docs/portfolio_jingbo_yan

Ann Uborevich-Borovskaya
2017《RobloX project_Bartlett M.AD RC4》:
https://issuu.com/annuborevich-borovskaya/docs/roblox_for_issuu

Chao Jiang
2017《INFINITE VOXELS| Portfolio |3D PRINTING |2016-2017》:
https://issuu.com/2273270/docs/infinitevoxels

Daniel Wigrig : http://www.danielwidrig.com/

Plethora : http://www.plethora-project.com/

Grasshopper3D : http://www.grasshopper3d.com/

Digital AEIOU : https://www.facebook.com/digitalaieou/

CCC Lab : https://labccc.wordpress.com/

Hall of Hexagons : http://www.drking.org.uk/hexagons/tess/housholder/index.html

Honeycomb (geometry) : https://en.wikipedia.org/wiki/Honeycomb_(geometry)

Alexander's Polyhedra : http://polyhedra.doskey.com/

GEOMETRY GARRET : http://schoengeometry.com/index.html

Ken Brakke Mathematics Department : http://facstaff.susqu.edu/brakke/

碎形 Fractal : http://www.atlas-zone.com/complex/fractals/

TESIGN Studio : https://tesignstudio.blogspot.com/

The Fermi Surface Database : http://www.phys.ufl.edu/fermisurface/

Guy's polyhedra pages : http://www.steelpillow.com/polyhedra/index.htm

Tiling Space with Regular and Semi-regular Polyhedra : 
http://met.iisc.ernet.in/~lord/webfiles/clusters/andreini.pdf