3D Parametric Study of Aloe Polyphylla

Sebastian Chu

FIBONACCI ALOE POLYPHYLLA: GOLDEN SPIRAL

0.0 Contents

1 .0

Biological Inspiration

2 .0

Initial Parametric Study

3 .0

Construction Sequence

4 .0

Parameter: Count

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.1 .2 .3 .4 .5 .6 .7

Second Parametric Study

Construction Sequence Continued

Parameter: Step Parameter: Max Height (Pedestals) Parameter: Max Height (Pedestals+Spires) Parameter: Reversed Max Height (Pedestals) Parameter: Reversed Max Height (Spires) Parameter: Attractor Location (Pedestals+Spires) Parameter: Attractor Location (Pedestals)

5 .0

Combined: Pedestals + Reversed Spires

6 .0

References + Animation

.1

.1

2

Geometric Approximations

Combined: Final Preference

Archistar Certificates

1.0 Biological Inspiration

1.1 Geometric Approximations

Background and Context

Golden Spiral

The Aloe polyphylla, also known as "Spiral Aloe," is a succulent plant which is naturally found in cold, mountainous habitats which often snow at high altitudes over 2,000 meters. It grows in crevices formed by rocks, or on slopes.

Overlaying a simple line drawing over the top view of the Spiral Aloe in Figure 5 clearly shows that the arrangement of the plant leaves is a close approximation of the golden spiral.

The golden spiral is a logarithmic spiral formed by a series of quarter circles scaling larger in size with each step. The proper scaling is an increase by the factor of φ each quarter circle.

The Spiral Aloe is unique due to the geometric configuration of its leaves, which oftentimes forms a near-perfect spiral geometry as it grows outward and matures.

In Figure 1, a top view of the Spiral Aloe illustrates the tighly compact spiral configuration of its leaves. Moreover, the leaves also appear to be smaller in the center, and scale larger toward the outer edges of the plant.

Figure 1. top view

It is difficult to model a true golden spiral, but there are several types of easier aproximations which are comparable to, and closely resemble the actual golden spiral.

Figure 2 provides a clearer view of the scaling effect. In addition to the thorny tip of each leaf, the side edges also host an array of smaller spikes which can be clearly seen in Figure 4.

Below are additional images of Aloe polyphylla.

In contrast, Figure 8 is a better approximation of the true golden spiral. The Fibonacci sequence is a sequence of numbers that starts with 1, and each subsequent number is the addition of the previous two numbers.

Using these numbers to form the side lengths of squares, the inscribed quarter circles within these squares resemble a golden spiral, and this will be the method of choice in studying the spiral aloe in this parametric study later on.

Figure 3 illustrates a full view of the whole plant, and it can be observed that the Spiral Aloe is not planar. Rather, it is somewhat spherical when fully grown.

The spiral configuration of its leaves can either rotate clockwise or counter-clockwise, but appears to conform to a specific pattern, one which will be studied in this project.

Figure 2. close-up perspective view

Figure 3. full perspective view 3

Figure 6. true golden spiral versus approximations

In Figure 7, the Lucas spiral is able to use a sequence of numbers similar to the Fibonacci numbers in order to approximate the golden spiral geometry. However, the Lucas spiral is less accurate when the numbers are smaller.

Figure 4. single leaf cross section view

Figure 7. the Lucas spiral approximation

Figure 5. overlaid spiral drawing

Figure 8. the Fibonacci spiral approximation 4

2.0 Initial Parametric Study

2.1 Second Parametric Study Step 5

First Interpretation

Step 4

Further Reinterpretation 298

The previous interpretation relied on mirroring and polar arraying spiral curves to achieve the approximate spiral of the Spiral Aloe. Much could be done with that interpretation, such as dividing points along the curves, then using perpendicular frames to orient leaves onto the spirals.

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With the understanding that there are several types of approximation for the logic of the golden spiral, a self interpretation was attempted.

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However, fundamentally that is a traslative approach using curves rather than a purely mathematical one.

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The second parametric study attempts to use the mathematics of sine and cosine to weave a point field that approximates the golden spiral. A voronoi diagram was generated using the point field as an input, creating much more freedom of configuration on a fundamental level.

Afterward, squares were overlaid on top of the self interpretation. These squares drawn with dashed red lines show that the self interpretation of the golden spiral is an accurate one, because it fits within the squares overlaid using the Fibonacci sequence.

Step 3

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Essentially, every even step the mirroring occurs on the y-axis, and every odd step the mirroring occurs on the x-axis.

This forms a close resemblance to the golden circle.

Step 2

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The experiment begins with a quarter circle of any length in step 1. In step 2, it is then mirrored along the y-axis and scaled up by 1.618 times the previous step. In step 3, the mirroring occurs about the x-axis instead.

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This allowed the study of the golden spiral to lead into different degrees and scales of spirals depending on the input, rather than relying on the initial mirrored spiral geometry like the previous study.

Step 4

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Figure 9. grasshopper first golden spiral attempt using curves

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Figure 10. grasshopper second golden spiral attempt using point fields

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3.0 Construction Sequence

1

Create a series of numbers with variable step & count. Solve each of the two expressions cos(t)*t and sin(t)*t with list of number from the series. Create points with x and y coordinates from the results of each expression, respectively.

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Set a custom point. Create attractor by finding the distance between each voronoi center point and the custom point. With the list of distance data, drive the vertical z-axis movement of the voronoi center points for the next step.

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3.1 Construction Sequence Continued

Configure the step size for the series of numbers until a desirable outcome is achieved. With a field of points computed from the series, generate a Voronoi diagram with variable radius and no boundary.

Remap the attractor values based on a variable min max limit.

Extrude the offset voronoi boundary surfaces in the z-axis direction based on these new values, to create a field of spiraling pedestals with varied heights.

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Generate negative (inward) offsets for each voronoi cell with variable distance to create visual separation. Get the area of each offset cell to extract the center point for the next series of computational exercises.

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6 Deconstruct brep to find the top surfaces of the pedestals. Using the attractor data, move the voronoi cell center points vertically above the pedestals. Extrude the top surface of pedestals to the center points above to generate spires.

Vertically extrude each offset voronoi cell in the z-axis direction to create a field of solid pedestals, separated by gaps in between due to the offsets.

Create a gradient with two shades of green color. Compute the total height of the pedestals + spires. Make a custom preview based on the heights. Shorter items are lighter shades of green, while taller items are darker shades of green.

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4.0 Parameter: Count

Isometric View

Isometric View

Top View

Variables

Variables

count = 30

step = 0.838

Unchanged Values

Unchanged Values

Variables

Variables

count = 65

step = 1.051

Unchanged Values

Unchanged Values

Variables

Variables

count = 100

step = 1.236

Unchanged Values

Unchanged Values

Variables

Variables

count = 145

step = 1.519

Unchanged Values

Unchanged Values

Variables

Variables

count = 186

step = 1.601

Unchanged Values

Unchanged Values

series start = 0 series step = 1.867 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

series start = 0 series step = 1.867 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

series start = 0 series step = 1.867 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

series start = 0 series step = 1.867 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

series start = 0 series step = 1.867 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

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4.1 Parameter: Step

Top View

series start = 0 series count = 186 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

series start = 0 series count = 186 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

series start = 0 series count = 186 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

series start = 0 series count = 186 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

series start = 0 series count = 186 min. height = 10 max. height = 100 cell radius = 57 offset = -1.5

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4.2 Parameter: Max Height (Pedestals)

Isometric View

4.3 Parameter: Max Height (Pedestals+Spires)

Isometric View

Top View

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

max. height = 50 series start = 0 series step = 1.640 count = 186 min. height = 1 cell radius = 57 offset = -1.5

max. height = 100 series start = 0 series step = 1.640 count = 186 min. height = 1 cell radius = 57 offset = -1.5

max. height = 150 series start = 0 series step = 1.640 count = 186 min. height = 1 cell radius = 57 offset = -1.5

max. height = 250 series start = 0 series step = 1.640 count = 186 min. height = 10 cell radius = 57 offset = -1.5

max. height = 350 series start = 0 series step = 1.640 count = 186 min. height = 1 cell radius = 57 offset = -1.5

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Top View

max. height = 10 series start = 0 series step = 1.601 count = 186 min. height = 10 cell radius = 57 offset = -1.5

max. height = 30 series start = 0 series step = 1.601 count = 186 min. height = 10 cell radius = 57 offset = -1.5

max. height = 60 series start = 0 series step = 1.601 count = 186 min. height = 10 cell radius = 57 offset = -1.5

max. height = 120 series start = 0 series step = 1.601 count = 186 min. height = 10 cell radius = 57 offset = -1.5

max. height = 200 series start = 0 series step = 1.601 count = 186 min. height = 10 cell radius = 57 offset = -1.5

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4.4 Parameter: Reversed Max Height (Pedestals)

Isometric View

4.5 Parameter: Reversed Max Height (Spires)

Isometric View

Top View

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

min. height = 10

series start = 0 series step = 1.519 count = 186 max. height = 10 cell radius = 57 offset = -1.5

min. height = 120

series start = 0 series step = 1.519 count = 186 max. height = 10 cell radius = 57 offset = -1.5

min. height = 250

series start = 0 series step = 1.519 count = 186 max. height = 10 cell radius = 57 offset = -1.5

min. height = 400

series start = 0 series step = 1.519 count = 186 max. height = 10 cell radius = 57 offset = -1.5

min. height = 600

series start = 0 series step = 1.519 count = 186 max. height = 10 cell radius = 57 offset = -1.5

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Top View

min. height = 10

series start = 0 series step = 1.783 count = 186 max. height = 1 cell radius = 57 offset = -1.5

min. height = 40

series start = 0 series step = 1.783 count = 186 max. height = 1 cell radius = 57 offset = -1.5

min. height = 100

series start = 0 series step = 1.783 count = 186 max. height = 1 cell radius = 57 offset = -1.5

min. height = 200

series start = 0 series step = 1.783 count = 186 max. height = 1 cell radius = 57 offset = -1.5

min. height = 400

series start = 0 series step = 1.783 count = 186 max. height = 1 cell radius = 57 offset = -1.5

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4.6 Parameter: Attractor Location (Pedestals+Spires)

Isometric View

4.7 Parameter: Attractor Location (Pedestals)

Isometric View

Top View

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

Variables

Variables

Unchanged Values

Unchanged Values

location = center

series step = 1.677 count = 186 min.height = 1 max. height = 200 cell radius = 57 offset = -1.5

location = north

series step = 1.677 count = 186 min.height = 1 max. height = 200 cell radius = 57 offset = -1.5

location = south

series step = 1.677 count = 186 min.height = 1 max. height = 200 cell radius = 57 offset = -1.5

location = east

series step = 1.677 count = 186 min.height = 1 max. height = 200 cell radius = 57 offset = -1.5

location = west

series step = 1.677 count = 186 min.height = 1 max. height = 200 cell radius = 57 offset = -1.5

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Top View

location = center

series step = 1.761 count = 186 min.height = 1 max. height = 350 cell radius = 57 offset = -1.5

location = north

series step = 1.761 count = 186 min.height = 1 max. height = 350 cell radius = 57 offset = -1.5

location = south

series step = 1.761 count = 186 min.height = 1 max. height = 350 cell radius = 57 offset = -1.5

location = east

series step = 1.761 count = 186 min.height = 1 max. height = 350 cell radius = 57 offset = -1.5

location = west

series step = 1.761 count = 186 min.height = 1 max. height = 350 cell radius = 57 offset = -1.5

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5.0 Combined: Pedestals + Reversed Spires

Isometric View

5.1 Combined: Final Preference

Top View

Variables

max height = 20

Unchanged Values point location = south series step = 1.761 count = 186 min.height = 1 cell radius = 57 offset = -1.5

Variables

max height = 80

Unchanged Values point location = south series step = 1.761 count = 186 min.height = 1 cell radius = 57 offset = -1.5

Variables

max height = 170

Unchanged Values point location = south series step = 1.761 count = 186 min.height = 1 cell radius = 57 offset = -1.5

Isometric View Variables

Variables

Unchanged Values

Unchanged Values

max height = 300

point location = south series step = 1.761 count = 186 min.height = 1 cell radius = 57 offset = -1.5

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Top View

location = center series step = 1.761 count = 186 min.height = 1 max. height = 350 cell radius = 57 offset = -1.5

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6.0 References and Animation

7.0 Archistar Certificates

References Figure 01.

Figure 02.

Figure 03.

Figure 04.

Figure 05.

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Figure 07.

Figure 08.

Figure 09.

Figure 10.

https://commons.wikimedia.org/wiki/File:Spiral_Aloe_detail.jpg http://www.tradewindsfruit.com/content/spiral-aloe.htm https://www.monrovia.com/plant-catalog/plants/1607/spiral-aloe/ https://depositphotos.com/stock-photos/aloe-vera-juice.html?search_params=eyJjdGYiOjF9 self generated image https://en.wikipedia.org/wiki/File:FakeRealLogSpiral.svg https://en.wikipedia.org/wiki/File:Lucas_number_spiral.svg https://en.wikipedia.org/wiki/File:FibonacciSpiral.svg self generated image self generated image

CERTIFICATE OF COMPLETION This certificate is awarded to

Sebastian Chu for completing the following course on 15t h of Sept ember 2020

Grasshopper Essentials Grade 91%

Passed

Attempt 1 Ben Coorey Director - ArchiStar Academy Architect #9413 RAIA | PhD, M Dig Arch, B Arch (Hons I), B Des (Arch/Dig Med)

Verify certificate.

The Ins titute of Dig ital De s ig n Aus tralian ( A.B.N 6 2143137115) Sydne y, Aus tralia

CERTIFICATE OF COMPLETION This certificate is awarded to

Sebastian Chu for completing the following course on 15t h of Sept ember 2020

Grasshopper Designer Grade 96%

Passed

Attempt 1 Ben Coorey Director - ArchiStar Academy Architect #9413 RAIA | PhD, M Dig Arch, B Arch (Hons I), B Des (Arch/Dig Med)

Verify certificate.

The Ins titute of Dig ital De s ig n Aus tralian ( A.B.N 6 2143137115) Sydne y, Aus tralia

Animation

https://youtu.be/XBsJLmHYmnU CERTIFICATE OF COMPLETION This certificate is awarded to

Sebastian Chu for completing the following course on 17t h of Sept ember 2020

Grasshopper Advanced Grade 89%

Passed

Attempt 1 Ben Coorey Director - ArchiStar Academy Architect #9413 RAIA | PhD, M Dig Arch, B Arch (Hons I), B Des (Arch/Dig Med)

Verify certificate.

The Ins titute of Dig ital De s ig n Aus tralian ( A.B.N 6 2143137115) Sydne y, Aus tralia

CERTIFICATE OF COMPLETION This certificate is awarded to

Sebastian Chu for completing the following course on 16t h of Sept ember 2020

Grasshopper Extensions Grade 90%

Passed

Attempt 1 Ben Coorey

Verify certificate.

Director - ArchiStar Academy Architect #9413 RAIA | PhD, M Dig Arch, B Arch (Hons I), B Des (Arch/Dig Med) The Ins titute of Dig ital De s ig n Aus tralian ( A.B.N 6 2143137115) Sydne y, Aus tralia

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