Difference between revisions of "ZCubes/Nature is Golden"

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==Video==
 
==Video==
 
<br/>
 
<br/>
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{{#ev:youtube|9ZzzN_dpbmI|480|left|Brocards Problem & Brown Numbers}}
+
{{#ev:youtube|9ZzzN_dpbmI|480|left|Nature is Golden}}
 
<br/><br/><br/><br/><br/><br/><br/><br/><br/><br/><br/><br/><br/>
 
<br/><br/><br/><br/><br/><br/><br/><br/><br/><br/><br/><br/><br/>
 
<br/>
 
<br/>
 +
 +
 +
==Code==
 +
<pre>
 +
1.FIBONNACI(100)
 +
 +
Output:
 +
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 .....
 +
</pre>
 +
 +
<pre>
 +
2. (1..1000)@["x*137.5","0.05*SQRT(x)"].toframe()
 +
 +
- When observed the above expression in charts, a spiral pattern is observed.
 +
</pre>
 +
 +
<pre>
 +
3. FIBONNACI(100)
 +
.chunks(2)
 +
.map(r=>r[1]/r[0])
 +
~
 +
 +
- Displays the golden ratio between two consecutive numbers of Fibonnaci series as shown below:
 +
 +
Infinity
 +
2
 +
1.6666666666666667
 +
1.625
 +
1.619047619047619
 +
1.6181818181818182
 +
1.6180555555555556
 +
1.6180371352785146
 +
1.618034447821682
 +
1.618034055727554
 +
1.6180339985218033
 +
1.6180339901755971
 +
1.618033988957902
 +
1.6180339887802426
 +
1.6180339887543225
 +
1.6180339887505408
 +
1.618033988749989
 +
1.6180339887499087
 +
1.618033988749897
 +
1.6180339887498951
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.6180339887498947
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.618033988749895
 +
1.6180339887498947
 +
1.618033988749895
 +
1.618033988749895
 +
1.6180339887498947
 +
1.618033988749895
 +
1.618033988749895
 +
...
 +
</pre>
 +
 +
<pre>
 +
4. ROOTS("x^2-x-1")
 +
 +
- Displays the quadratic roots as:
 +
[1/2−1/2∗5(1/2),1/2+1/2∗5(1/2)]
 +
 +
 +
(1+SQRT(5))/2
 +
 +
- Displays golden ratio as 1.618033988749895
 +
</pre>
 +
 +
<pre>
 +
5. Golden Angle:
 +
 +
gr=[(1+SQRT(5))/2,(1-SQRT(5))/2]
 +
gr[0]
 +
 +
Displays the golden ratio: 1.618033988749895
 +
 +
 +
360*(1-1/gr[0])
 +
 +
Displays the golden angle: 137.50776405003788
 +
 +
 +
360*(1-1/gr[0])<>deg<>rad
 +
 +
Displays the golden angle in radians: 2.3999632297286535rad
 +
 +
 +
</pre>
 
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*[[Z^3 Language Documentation]]
 
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Latest revision as of 09:48, 21 September 2021

Nature is Golden


The beautiful golden sunflowers are a mathematical marvel. The pattern of seeds in a sunflower follows the Fibonacci series. This video demonstrates how nature uses Fibonacci series, golden ratio and golden angle in sunflower, pinecones etc, and secrets behind our admiration for certain designs, architecture and art. Enjoy watching!

ZCubes is an omnifunctional computing platform. You can create documents, spreadsheets, charts & presentations, draw, paint, graphics & animations, web pages and much more with ZCubes.

Video


Nature is Golden
















Code

1.FIBONNACI(100)

Output: 
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181	6765 10946 17711 28657 46368 75025 .....

2. (1..1000)@["x*137.5","0.05*SQRT(x)"].toframe()

- When observed the above expression in charts, a spiral pattern is observed.

3. FIBONNACI(100)
	.chunks(2)
	.map(r=>r[1]/r[0])
~

- Displays the golden ratio between two consecutive numbers of Fibonnaci series as shown below:

Infinity
2
1.6666666666666667
1.625
1.619047619047619
1.6181818181818182
1.6180555555555556
1.6180371352785146
1.618034447821682
1.618034055727554
1.6180339985218033
1.6180339901755971
1.618033988957902
1.6180339887802426
1.6180339887543225
1.6180339887505408
1.618033988749989
1.6180339887499087
1.618033988749897
1.6180339887498951
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.6180339887498947
1.618033988749895
1.618033988749895
1.618033988749895
1.618033988749895
1.6180339887498947
1.618033988749895
1.618033988749895
1.6180339887498947
1.618033988749895
1.618033988749895
...

4. ROOTS("x^2-x-1")

- Displays the quadratic roots as: 
[1/2−1/2∗5(1/2),1/2+1/2∗5(1/2)]


(1+SQRT(5))/2 

- Displays golden ratio as 1.618033988749895

5. Golden Angle:

gr=[(1+SQRT(5))/2,(1-SQRT(5))/2]
gr[0]

Displays the golden ratio: 1.618033988749895


360*(1-1/gr[0])

Displays the golden angle: 137.50776405003788


360*(1-1/gr[0])<>deg<>rad

Displays the golden angle in radians: 2.3999632297286535rad



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