seashell interpretation in architectural formsmypages.iit.edu/~krawczyk/shell01/curve_01.pdf ·...

Seashell Interpretation in Architectural Forms

From the study of seashell geometry, the mathematical process onshell modeling is extremely useful. With a few parameters,families of coiled shells were created with endless variations.This unique process originally created to aid the study ofpossible shell forms in zoology. The similar process can bedeveloped to utilize the benefit of creating variation of thepossible forms in architectural design process.

The mathematical model for generating architectural formsdeveloped from the mathematical model of gastropods shell,however, exists in different environment. Normally humanarchitectures are built on ground some are partly undergroundsubjected different load conditions such as gravity load, windload and snow load.

In human architectures, logarithmic spiral is not necessarybecome significant to their shapes as it is in seashells. Sincearchitecture not grows as appear in shells, any mathematicalcurve can be used to replace this, so call, growth spiral thatoccur in the shells during the process of growth. The generatingcurve in the shell mathematical model can be replaced withdifferent mathematical closed curve. The rate of increase in thegenerating curve needs not only to be increase. It can beincrease, decrease, combination between the two, and constant byignore this parameter. The freedom, when dealing witharchitectural models, has opened up much architectural solution.In opposite, the translation along the axis in shell model isgreatly limited in architecture, since it involves the problemof gravity load and support condition.

Mathematical Curves

To investigate the possible architectural forms base on the ideaof shell mathematical models, the mathematical curves arereviewed. As in shell curve, the study in this step will focuson utilizing every mathematical curves property. Most influentworks reviewed in this area are A Handbook on Curves and TheirProperties by Robert C. Yates 1947, A Book of Curves by EdwardHarrington Lockwood 1961, A Catalog of Special Plane Curves byJ. Dennis Lawrence 1972, Concise Encyclopedia of Mathematics(CD-ROM) by Eric W. Weisstein 1999, and some related woks suchas Wobbly Spiral by John Shape and mathematical related website.

In this chapter, all mathematical curves are divided into fourcurve groups in order to facilitate a further development oftheir property.

1. Close Curves

The mathematical plane curve that always complete or closeitself at the 360-degree. Close curves are the followings:

• Circle • Hypocycloid• Ellipse • Epitrochoid• Piriform • Hypotrochoid• Bicorn • Cardioid• Astroid • Nephroid of Freeths• Deltoid • Cayley’s Sextic• Eight • Lissajous• Bifolium • Nephroid• Folia • Rhodonea• Lemniscate of Bernoulli • Limacon of Pascal• Epicycloid • Hippopede

2. Open Curves

The mathematical plane curve that never complete or closeitself. It represents a single body of curve in one loop (360-degree). This group is divided into three minor group bases onthe curve characteristic.

2.1 Increasing Curves

This type of open curve creates only a single curvature withoutrepeating. Increasing curves are the followings:

• Hyperbola • Cross• Parabola • Conchoid of Nicomedes• Pedals of Parabola • Tschirnhausen’s Cubic• Semi-Cubical Parabola • Witch of Agnesi• Catenary • Cissoid of Diocles• Kappa • Right Strophoid• Kampyle of Eudoxus • Trisectrix of Maclaurin• Bullet Nose • Folium of Descartes• Serpentine • Tractrix

2.2 Periodic Curves

In opposite of the increasing curve, this type of open curvecontinuously repeats similar curvature along the increasingangle. Periodic curves are the followings:

• Sine Curve • Cycloid• Cosind Curve • Trochoid

2.1 Spiral Curves

Spiral is open curve that every point in the curve travelsaround the center axis or pole. Spiral curves are thefollowings:

• Involute of a Circle • Logarithmic Spiral• Fermat’s Spiral • Hyperbolic Spiral• Poinsot’s Spirals • Lituus• Cochleoid • Euler’s Spiral• Archimedes’ Spiral

2.2 Special Spiral Curves

These curves are developed from others mathematical curves knownfor mathematician but interpret them into new mathematicalproperty. Special Spiral curves are the followings:

• Wobbly Spiral • Rectangular Spiral

3. Three-dimensional Curves

This curve is either close or open. The major difference is thatthe curve takes the height in z-axis while the others are notconsidering this axis. Three-dimensional curves are thefollowings:

• Spherical spiral • Helix• Cylinder

Each mathematical function and its property mentioned above canbe found in a greater detail in appendix.

Seashell Properties and Architecture Properties

Seashell Properties

Again, base on the study in the previous chapter, these areknown parameters in shell mathematical study and modeling. Forfurther study of this research, however, a short terminology isintroduced to represent these four parameters originally calledafter those researchers in biology and zoology. These new termsare called as the followings:

Biology and ZoologyTerms

ResearchTerms

1. The shape of generating curve. = Section2. The rate of increase of the generating curve. = Growth3. The position of generating curve to the axis. = Path4. The rate translation along the axis. = Height

Figure xx: SectionA seashell cross sectionaloutline of the hollow shelltube.

Figure xx: GrowthThe rate of increase of thesection in size.

Figure xx: PathThe position of generatingcurve to the axis.

Figure xx: HeightThe rate translation along theaxis in vertical direction.

Mathematical Curves Properties

A unique property in each parameter can be explored with variousmathematical properties. In seashell these relationships arelimited depend upon the actual geometry of shells. Here, thelimitations are less. The link from seashell properties tomathematical properties is presented in the following table.

ResearchTerms

Seashell Relative to ItsMathematical Properties

MathematicalCurves Properties

1. Section = Circle, Ellipse, Free shape = Close Curves2. Growth = Exponential Function = Open Curves3. Path = Logarithmic Spiral = Open, 3D Curve4. Height = Progressive Function = Open Curves

Architectural Properties

In architectural forms, there are some others parameters thatarchitects and engineers have to take into consideration. Thesearchitectural parameters came from basic design principle inarchitecture or the fundamental functions and conditions thatmake architectural forms work. In this research, the goal isexperimenting on architectural forms in general. It is notintended to find architectural form to suite one particularproject. Without a specific requirement of the actual site andactual project, the architectural parameters can be set ingeneral as the followings:

1. Ground condition.2. Human scale and proportion.3. Wall opening and circulation.4. Support condition.

Integrating these three unique properties, which are seashell,mathematics, and architecture, certainly yields the result inthe mathematical interpretation of architectural forms thatmaintain the seashell signature. The integration of these threecan be done by programming language, similar process in chapterIII.

The parameters involved in the program will start from less to acomplex one. The mathematical functions behind thesearchitectural forms are reviewed as follows:

Circle:

Mathematical Function:

x = r cos t

y = r sin t , - � t �

r = 1, 1.5, 2

Factors: r = radius

Reference:J. Dennis Lawrence, A Catalog of Special Plane Curves, DoverPublications, Inc., 1972 (p. 65)

Ellipse:


x = a cos t

y = b sin t , - � t �

a = 1, b = 2 a = 2, b = 1

Factors: a = radius in x-axisb = radius in y-axis


Piriform: (De Longchamps, 1886)

The piriform is also known as the pear-shaped quartic.


x = a (1 + sin t)3

y = b cos t (1 + sin t) , -2

� t �2

a = 1, 2, 3 a = 1b = 1 b = 1, 2, 3

Factors: a = x-axis distanceb = y-axis distance


Bicorn: (Sylvester, 1864)


x = a sin t

a cos2 t (2 + cos t)y =

3 + sin2 t , - � t �

a = 1, 2, 3

Factors: a = scale of curve


Epicycloid: (Roemer, 1674)


mx = m cos t – b cos

bt

my = m sin t – b sin

bt , - � � t � �

m = 1; b = 0.1 m = 1; b = 0.2

m = 1; b = 0.25 m = 1; b = 0.5

Factors: m = scale of curveb = number of cusps


Epitrochoid:


mx = m cos t – h cos

bt

my = m sin t – h sin

bt , - � � t � �

m = 1; b = 0.1; h = 0.2 m = 1; b = 0.2; h = 0.2

m = 1; b = 0.2; h = 0.1 m = 1; b = 0.2; h = 0.5

Factors: m = scale of curveb = number of cuspsh = overlapping


Hypotrochoid:


nx = n cos t + h cos

bt

ny = n sin t – h sin

bt , - � � t � �

n = 1; b = 0.1; h = 0.2 n = 1; b = 0.2; h = 0.2

n = 1; b = 0.2; h = 0.1 n = 1; b = 0.2; h = 0.5

Factors: n = scale of curveb = number of cuspsh = overlapping


Eight:


x = a cos t

y = a sin t cos t , - � � t � �

a = 1, 2, 3



Lemniscate of Bernoulli:


a cos tx =

1 + sin2 ta sin t cos t

y =1 + sin2 t , - � � t � �

a = 1, 2, 3



Astroid: (Roemer, 1674; Bernolli, 1691)


x = a (3 cos t + cos 3 t)

y = a (3 sin t – sin 3 t) , - � � t � �

a = 1, 2, 3



Hypocycloid:


nx = n cos t + b cos

bt

ny = n sin t – b sin

bt , - � � t � �

n = 1; b = 0.1 n = 1; b = 0.2

n = 1; b = 0.25 n = 1; b = 0.5

Factors: n = scale of curveb = number of cusps


Cardioid: (Koersma, 1689)


x = 2 a cos t (1 + cos t)

y = 2 a sin t (1 + cos t) , - � � t � �

a = 0.5, 0.75, 1



Nephroid of Freeths:


1r = a (1 + 2 sin

2 θ)

x = r cos t

y = r sin t , - � � t � �

a = 0.75 a = 1



Cayley’s Sextic: (Maclaurin, 1718)


1r = a cos3

3θ

x = r cos t

y = r sin t , - � � t � �

a = 0.75 a = 1



Bifolium:


r = 4 a sin2 θ cos θ

x = r cos t

y = r sin t , - � � t � �

a = 1, 2, 3


Reference:Eric W. Weisstein, Concise Encyclopedia of Mathematics, CRCPress, CD-ROM Edition, 1999

Folia: (Kepler, 1609)


r = cos θ (4 a sin2 θ - b)

x = r cos t

y = r sin t , - � � t � �

a = 1, 2, 3; b = 1, 2, 3 a = 1, b = 4

a = 3, b = 2 a = 2, b = 4

Factors: a = scale of curveb 4 a, Single foliumb = 0, Bifolium0�b�4 a, Trifolium


Lissajous:


x = a sin (n t + c)

y = b sin t , - � � t � �

a = 1, b = 1, c = 2, n = 4 a = 1, b = 1, c = 4, n = 4

a = 1, b = 2, c = 2, n = 4 a = 2, b = 1, c = 2, n = 4

Factors: a = x-axis distanceb = y-axis distancec = x-axis factorn = number of loop


Nephroid: (Huygens, 1678)


x = a (3 cos t – cos 3 t)

y = a (3 sin t – sin 3 t) , - � � t � �

a = 0.5, 0.75



Rhodonea: (Grandi, 1723)


r = a cos m θ

x = r cos t

y = r sin t , - � � t � �

a = 1, m = 3 a = 2, m = 4

Factors: a = scale of curvem = number of loop


Limacon of Pascal: (Pascal, 1650)


r = 2 a cos θ + b

x = r cos t

y = r sin t , - � � t � �

a = 0, b = 1 a = 1, b =1

a = 1, b = 2 a = 1, b = 3

Factors: a = scale of curve; a = 0, Circleb = a, Trisectrixb = 2 a, Cardioid


Deltoid: (Euler, 1745)


x = a (2 cos t + cos 2 t)

y = a (2 sin t – sin 2 t) , - � � t � �

a = 0.5, 0.75, 1



Hippopede: (Proclus, ca. 75 B.C.)


x = 2 cos t sqrt(a b – b2 sin2 t)

y = 2 sin t sqrt(a b – b2 sin2 t) , - � � t � �

a = 1, b = 1 a = 1, b = 0.9

a = 1, b = 0.5 a = 1, b = 0.1

Factors: a = scale of curveb < a


Hyperbola:


x = a sec t

y = b tan t , - � � t � �

a = 1, b = 1 a =2, b = 1



Kappa: (Gutschoven’s curve)


x = a cos t cot t

y = a cos t , 0 < t < 2�

a = 1, 2, 3



Kampyle of Eudoxus:


x = a sec t� 3�

y = a tan t sec t , -2

< t <2

a = 1, 2, 3



Bullet Nose: (Schoute, 1885)


x = a cos t

y = b cot t , - � < t < �

a = 1, b = 1 a = 3, b = 1



Cross:


x = a sec t

y = b csc t , - � < t < �

a = 1, b = 1



Conchoid of Nicomedes: (Nicomedes, 225 B.C.)


x = b + a cos t� 3�

y = tan t (b + a cos t) , -2

< t <2

a = 1, b = 1



Parabola:

Mathematical function:

x = a t2

y = 2 a t , - � < t < �

a = 1, 2, 3



Semi-Cubical Parabola: (Isochrone)


x = 3 a t2

y = 2 a t3 , - � < t < �

a = 1, 2, 3



Tschirnhausen’s Cubic:


x = a (1 – 3 t2)

y = a t (3 – t2) , - � < t < �

a = 0.25, 0.5



Witch of Agnesi: (Fermat, 1666; Agnesi, 1748)


x = 2 a tan t� �

y = 2 a cos2 t , -2

< t <2

a = 0.5, 0.75, 1



Pedals of Parabola:


x = a (sin2 t – m cos2 t)� �

y = a tan t (sin2 t – m cos2 t) , -2

< t <2

a = 1; m = 1, 2, 3 a = 1; b = -0.2, -0.4, -0.6

Factors: a = scale of curvem > 0, one loopm � 0, no loop


Cissoid of Diocles: (Diocles, ca. 200 B.C.)


x = a sin2 t

y = a tan t sin2 t , - � < t < �

a = 1, 2, 3



Right Strophoid: (Barrow, 1670)


x = a (1 – 2 cos2 t)� �

y = a tan t (1 – 2 cos2 t) , -2

< t <2

a = 1 a = 2



Trisectrix of Maclaurin: (Maclaurin, 1742)


x = a (1 – 4 cos2 t)� �

y = a tan t (1 – 4 cos2 t) , -2

< t <2

a = 0.5, 0.75, 1



Folium of Descartes: (Descartes, 1638)


3 a tx =

1 + t3

3 a t2y =

1 + t3, - � < t < �

a = 0.5, 0.75, 1



Serpentine:


x = a cot t

y = b sin t cos t , 0 < t < �

a = 1, b =1 a = 1, b = 2



Tractrix: (Huygens, 1692)


x = a ln (sec t + tan t) – a sin t� �

y = a cos t , -2

< t <2

a = 1, 2, 3



Catenary:


x = t

ty = a cosh

a, - � < t < �

a = 1



Involute of a Circle: (Huygens, 1693)


x = a (cos t + t sin t)

y = a (sin t – t cos t) , - � < t < �

a = 0.5; 0 – 360 degree



Poinsot's spirals:


a ar =

cosh n θand

sinh n θx = r cos t

y = r sin t , 0 < t < �

(a / cosh n θ) (a / sinh n θ)

a = 1, b = 4 a = 1, b = 4a = 1, b = 1 a = 1, b = 1a = 4, b = 1 a = 4, b = 1

Factors: a = x-axis distanceB = y-axis distance


Fermat’s Spiral:


r = a sqrt θ

x = r cos t

y = r sin t , 0 < t < �

a = 1; 0 – 360 degree



Cochleoid: (Bernoulli, 1726)


a sin θr =

θx = r cos t

y = r sin t , 0 < t < �

a = 4; 0 – 1080 degree



Logarithmic: (Descartes, 1638)


r = exp ( a θ )

x = r cos t

y = r sin t , 0 < t < �

(Missing log’s image)



Archimedes’ Spiral: (Sacchi, 1854)


r = a θ

x = r cos t

y = r sin t , 0 < t < �

a = 0.2; 0 – 1080 degree



Hyperbolic Spiral:


ar = θx = r cos t

y = r sin t , 0 < t < �

a = 3; 0 – 1080 degree



Lituus:

Mathemaical Function:

ar = √θx = r cos t

y = r sin t , 0 < t < �

a = 1; 0 – 1080 degree



Wobbly Spiral:


4θr = (1 + 2 k sin3

) a eθk

ln –�k =

32

; – = 270

x = r cos t

y = r sin t , 0 < t < �

a = 0.5; 0 – 1440 degree


Reference:John Sharp, Mathematics in School, November 1997

Euler’s Spiral: (Euler, 1744)


sin tx = a √t d t

cos ty = a

√td t , 0 � t < �

a = 3, d = 1

Factors: a = scale of curved = scale of curve


Cycloid: (Mersenne, 1599)


x = a t + h sin t

y = a – h cos t , -� < t < �

a = 0.3, h = 0.1 a = 0.3, h = 0.3

a = 0.3, h = 0.5 a = 0.3, h = 2

Factors: a = scale of curveh = overlapping0 – 1080 degree


Trochoid:


x = a t – h sin t

y = a – h cos t , -� < t < �

a = 0.3, h = 0.5 a = 0.3, h = 1

a = 0.3, h = 2

Factors: a = scale of curveh = overlapping0 – 1080 degree


Sine and Cosine Curve:


x = x

y = X sin t , 0 < t < �

(Missing sine and cosine images)

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