Golden_Ratio_Presentation_Pavel_Khokhlov

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THE GOLDEN RATIO
A simple idea connecting mathematics, nature and art
Based on “Maths in a minute: The golden ratio”
Article author: Plus Maths editorial team
Presented by Pavel Khokhlov

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1. What is the golden ratio?
• The golden ratio is a special relationship between two
lengths.
A+B
• The ratio of the whole length to the longer part equals
the ratio of the longer part to the shorter part.
• Its value is approximately 1.618.
Longer part (A)
Shorter (B)
Whole / longer = longer / shorter
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2. Where does the number come from?
• Let the longer part be A and the shorter part be B.
• The defining relationship is (A + B) / A = A / B.
• If we call this ratio φ (phi), then φ² = φ + 1.
• Solving gives φ = (1 + √5) / 2 ≈ 1.618.
φ² = φ + 1
φ ≈ 1.6180339…
An irrational number: its decimal never
ends or repeats.
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3. The Fibonacci sequence
• Start with 1 and 1.
• Each new number is the sum of the two numbers before it.
• The sequence begins: 1, 1, 2, 3, 5, 8, 13, 21, 34…
• Ratios of neighbouring numbers get closer to the golden
ratio.
1
1
2
3
5
Add the previous two numbers
8 ÷ 5 = 1.6
13 ÷ 8 = 1.625
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4. Mathematics in nature
• Fibonacci-like spiral counts can be observed in some
sunflowers and pinecones.
• These patterns are connected with how plants arrange
seeds and scales as they grow.
• Mathematics helps us describe patterns; nature does not
always follow a perfect formula.
Seed patterns can form spirals
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5. Art, design and architecture
• The golden ratio has interested artists, architects and
designers for centuries.
• It can be used as one way to plan proportions in a picture or
a layout.
• Claims that famous artworks or buildings always use this
ratio should be checked carefully.
Golden rectangle:
width / height ≈ 1.618
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6. Why is it interesting?
• One number connects geometry, algebra and number
patterns.
• The same ratio appears in mathematical models of some
natural growth patterns.
• It shows how a simple rule can create surprising
connections.
GEOMETRY
NUMBER PATTERNS
REAL WORLD
Ratios and
rectangles
Fibonacci
sequence
Nature and
design
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7. Source and key takeaway
Main article
“Maths in a minute: The golden ratio” — Plus Maths, 9 February 2017
https://plus.maths.org/maths-minute-golden-ratio
Key takeaway
The golden ratio is not magic. It is a precise mathematical relationship that helps us
explore number patterns, geometry and some forms found in nature.
This presentation summarizes the article in simpler English.
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THANK YOU
for your attention
Presented by Pavel Khokhlov