{"id":4800,"date":"2023-06-21T15:51:21","date_gmt":"2023-06-21T15:51:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=4800"},"modified":"2025-08-28T03:39:12","modified_gmt":"2025-08-28T03:39:12","slug":"proportion-and-the-golden-ratio-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/proportion-and-the-golden-ratio-learn-it-2\/","title":{"raw":"Proportion and the Golden Ratio: Learn It 2","rendered":"Proportion and the Golden Ratio: Learn It 2"},"content":{"raw":"

Proportional Systems<\/h2>\r\n

Joining a cake decorating course will introduce you to crafting icing flowers, where you\u2019ll learn that flowers, depending on their size, typically require groupings of [latex]5[\/latex], [latex]8[\/latex], or [latex]13[\/latex] petals for a realistic look.<\/p>\r\n

If learning to draw portraits, you may be surprised to learn that eyes are approximately halfway between the top of a person\u2019s head and their chin.<\/p>\r\n

Studying architecture, we find examples of buildings that contain golden rectangles and ratios that add to the beautifying of the design. The Parthenon (seen below), which was built around 400 BC, as well as modern-day structures such the Washington Monument are two examples containing these relationships.<\/p>\r\n

\r\n[caption id=\"attachment_4804\" align=\"aligncenter\" width=\"717\"]\"Image Figure 1. Credit: \u201cParth\u00e9non\u201d by Julien Maury\/Flickr, Public Domain Mark 1.0[\/caption]\r\n<\/center>
<\/center>\r\n

 <\/p>\r\n

These seemingly unrelated examples and many more highlight mathematical relationships that we associate with beauty in artistic form. These include the \"Rule of Thirds\" and \" the \"Golden Ratio\".<\/p>\r\n

Golden Ratio<\/h3>\r\n

The golden ratio<\/strong>, also known as the golden proportion or the divine proportion, is a ratio aspect that can be found in beauty from nature to human anatomy as well as in golden rectangles that are commonly found in building structures.<\/p>\r\n

The golden ratio is expressed in nature from plants to creatures such as starfish, honeybees, seashells, and more. It is commonly noted by the Greek letter [latex]\\phi[\/latex], pronounced \"fee\".<\/p>\r\n

\r\n

[latex]\\phi=\\frac{1+\\sqrt{5}}{2}[\/latex], which has a decimal value approximately equal to [latex]1.618[\/latex].<\/p>\r\n<\/section>\r\n

\r\n
\r\n

golden ratio<\/h3>\r\n\r\nOften represented by the Greek letter Phi ([latex]\\phi[\/latex]), the Golden Ratio<\/strong> (approximately [latex]1:1.618[\/latex]) has been used in art and architecture to create aesthetically pleasing proportions.\u00a0<\/div>\r\n<\/section>\r\n

The golden ratio has been used by artists through the years and can be found in art dating back to 3000 BC. Leonardo da Vinci is considered one of the artists who mastered the mathematics of the golden ratio, which is prevalent in his artwork such as Virtuvian Man (seen below). This famous masterpiece highlights the golden ratio in the proportions of an ideal body shape.<\/p>\r\n

\r\n[caption id=\"attachment_4809\" align=\"aligncenter\" width=\"588\"]\"Vitruvian Figure 2. The Virtuvian Man by Leonardo da Vinci[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

The golden ratio is approximated in several physical measurements of the human body and parts exhibiting the golden ratio are simply called golden. The ratio of a person\u2019s height to the length from their belly button to the floor is [latex]\\phi[\/latex] or approximately [latex]1.618[\/latex]. The bones in our fingers (excluding the thumb), are golden as they form a ratio that approximates [latex]\\phi[\/latex]. The human face also includes several ratios and those faces that are considered attractive commonly exhibit golden ratios.<\/p>\r\n

If a person\u2019s height is [latex]5[\/latex] ft [latex]6[\/latex] in, what is the approximate length from their belly button to the floor rounded to the nearest inch, assuming the ratio is golden? [reveal-answer q=\"160930\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160930\"] Step 1<\/strong>: Convert the height to inches: [latex]5\\text{ ft } 6\\text{ in }=66\\text{ in }[\/latex]\r\n\r\n

 <\/p>\r\nStep 2<\/strong>: Calculate the length from the belly button to the floor, [latex]L[\/latex].[latex]\\frac{66}{L} = 1.618[\/latex][latex]L = 40.8 \\text{ in }[\/latex] The length from the person\u2019s belly button to the floor would be approximately [latex]41[\/latex] in. [\/hidden-answer]<\/section>\r\n

The golden ratio ratio is derived from the Fibonacci sequence.\u00a0<\/p>\r\n

Fibonacci Sequence and Application to Nature<\/h4>\r\n

The Fibonacci sequence<\/strong> can be found occurring naturally in a wide array of elements in our environment from the number of petals on a rose flower to the spirals on a pine cone to the spines on a head of lettuce and more.<\/p>\r\n

The Fibonacci sequence can be found in artistic renderings of nature to develop aesthetically pleasing and realistic artistic creations such as in sculptures, paintings, landscape, building design, and more. It is the sequence of numbers beginning with [latex]1[\/latex], [latex]1[\/latex], and each subsequent term is the sum of the previous two terms in the sequence ([latex]1, 1, 2, 3, 5, 8, 13, \u2026[\/latex]).<\/p>\r\n

\r\n[caption id=\"attachment_4815\" align=\"aligncenter\" width=\"445\"]\"Rose Figure 3. Rose petals appear in a Fibonacci spiral[\/caption]\r\n<\/center>
<\/center>\r\n

 <\/p>\r\n

\r\n
\r\n

Fibonacci sequence<\/h3>\r\n\r\nThe Fibonacci sequence<\/strong> is a series of numbers in which each number is the sum of the two preceding ones. Typically starting with [latex]0[\/latex] and [latex]1[\/latex], the sequence proceeds as follows: [latex]0, 1, 1, 2, 3, 5, 8, 13,[\/latex] and so on.<\/div>\r\n<\/section>\r\n
\r\n

The petal counts on some flowers are represented in the Fibonacci sequence. A daisy is sometimes associated with plucking petals to answer the question \u201cThey love me, they love me not.\u201d Interestingly, a daisy found growing wild typically contains [latex]13[\/latex], [latex]21[\/latex], or [latex]34[\/latex] petals and it is noted that these numbers are part of the Fibonacci sequence. The number of petals aligns with the spirals in the flower family.<\/span><\/p>\r\n<\/section>\r\n

[videopicker divId=\"tnh-video-picker\" title=\"Fibonacci sequence\" label=\"Select Video\"] [videooption displayName=\"Understanding the Fibonacci Sequence in Everyday Life\" value=\"https:\/\/youtu.be\/2tv6Ej6JVho\"][videooption displayName=\"The Fibonacci Sequence: Nature's Code\" value=\"https:\/\/youtu.be\/wTlw7fNcO-0\"] [videooption displayName=\"The magic of Fibonacci numbers - Arthur Benjamin\" value=\"https:\/\/youtu.be\/SjSHVDfXHQ4\"] [\/videopicker]\r\n\r\n

 <\/p>\r\n

You can view the transcript for \u201cWhat is the Fibonacci Sequence & the Golden Ratio? Simple Explanation and Examples in Everyday Life\u201d here (opens in new window).<\/a><\/p>\r\n

You can view the transcript for \u201cThe Fibonacci Sequence: Nature's Code\u201d here (opens in new window).<\/a><\/p>\r\n

You can view the transcript for \u201cThe magic of Fibonacci numbers | Arthur Benjamin | TED\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n

Suppose you were creating a rose out of icing, assuming a Fibonacci sequence in the petals, how many petals would be in the row following a row containing [latex]13[\/latex] petals? [reveal-answer q=\"160931\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160931\"] The number of petals on a rose is often modeled with the numbers in the Fibonacci sequence, which is [latex]1, 1, 2, 3, 5, 8, 13,\u2026,[\/latex] where the next number in the sequence is the sum of [latex]8+13=21[\/latex]. There would be [latex]21[\/latex] petals on the next row of the icing rose. [\/hidden-answer]<\/section>\r\n
[ohm2_question hide_question_numbers=1]8923[\/ohm2_question]<\/section>\r\n

Golden Ratio and the Fibonacci Sequence Relationship<\/h4>\r\n

Mathematicians for years have explored patterns and applications to the world around us and continue to do so today. One such pattern can be found in ratios of two adjacent terms of the Fibonacci sequence.\u00a0 When computing the ratio of the larger number to the preceding number such as [latex]\\frac{8}{5}[\/latex] or [latex]\\frac{13}{8}[\/latex], it is fascinating to find the golden ratio emerge. As larger numbers from the Fibonacci sequence are utilized in the ratio, the value more closely approaches [latex]\\phi[\/latex], the golden ratio.<\/p>\r\n

The [latex]24[\/latex]th Fibonacci number is [latex]46,368[\/latex] and the [latex]25[\/latex]th is [latex]75,025[\/latex]. Show that the ratio of the [latex]25[\/latex]th and [latex]24[\/latex]th Fibonacci numbers is approximately [latex]\\phi[\/latex]. Round your answer to the nearest thousandth. [reveal-answer q=\"160932\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160932\"] [latex]\\frac{75,025}{46,368}=1.618[\/latex]; The ratio of the [latex]25[\/latex]th and [latex]24[\/latex]th term is approximately equal to the value of [latex]\\phi[\/latex] rounded to the nearest thousandth, [latex]1.618[\/latex]. [\/hidden-answer]<\/section>\r\n

Golden Rectangles<\/h3>\r\n

Turning our attention to man-made elements, the golden ratio can be found in architecture and artwork dating back to the ancient pyramids in Egypt to modern-day buildings such as the UN headquarters.<\/p>\r\n

\r\n[caption id=\"attachment_4827\" align=\"aligncenter\" width=\"652\"]\"The Figure 4. The pyramids of Giza in Egypt[\/caption]\r\n<\/center>\r\n

The ancient Greeks used golden rectangles<\/strong> \u2014 any rectangles where the ratio of the length to the width is the golden ratio \u2014 to create aesthetically pleasing as well as solid structures, with examples of the golden rectangle often being used multiple times in the same building such as the Parthenon. Golden rectangles can be found in twentieth-century buildings as well, such as the Washington Monument.<\/p>\r\n

\r\n
\r\n

golden rectangle<\/h3>\r\n\r\nA golden rectangle<\/strong> is a rectangle in which the ratio of the length to the width is the golden ratio, approximately [latex]1.618[\/latex]. If you have a rectangle where the longer side (length) is [latex]a[\/latex] and the shorter side (width) is [latex]b[\/latex], it will be a Golden Rectangle when the ratio [latex]a\/b[\/latex] is equal to [latex](a+b)\/a[\/latex], which equals the golden ratio.<\/div>\r\n<\/section>\r\n

One fascinating property of a Golden Rectangle is that if you remove a square from the rectangle (a square whose sides are equal to the shortest length of the rectangle), the rectangle that's left over will also be a Golden Rectangle. This property leads to the creation of a logarithmic spiral, which can be drawn within the rectangle by connecting points within the subdivided squares. This spiral is often found in nature, for example, in the arrangement of seeds in sunflowers and the shape of certain galaxies.<\/p>\r\n

A frame has dimensions of [latex]8[\/latex] in by [latex]6[\/latex] in. Calculate the ratio of the sides rounded to the nearest thousandth and determine if the size approximates a golden rectangle. [reveal-answer q=\"160933\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160933\"] [latex]\\frac{8}{6} = 1.333[\/latex]; A golden rectangle\u2019s ratio is approximately [latex]1.618[\/latex]. The frame dimensions are close to a golden rectangle. [\/hidden-answer]<\/section>\r\n
[ohm2_question hide_question_numbers=1]8924[\/ohm2_question]<\/section>","rendered":"

Proportional Systems<\/h2>\n

Joining a cake decorating course will introduce you to crafting icing flowers, where you\u2019ll learn that flowers, depending on their size, typically require groupings of [latex]5[\/latex], [latex]8[\/latex], or [latex]13[\/latex] petals for a realistic look.<\/p>\n

If learning to draw portraits, you may be surprised to learn that eyes are approximately halfway between the top of a person\u2019s head and their chin.<\/p>\n

Studying architecture, we find examples of buildings that contain golden rectangles and ratios that add to the beautifying of the design. The Parthenon (seen below), which was built around 400 BC, as well as modern-day structures such the Washington Monument are two examples containing these relationships.<\/p>\n

\n
\"Image
Figure 1. Credit: \u201cParth\u00e9non\u201d by Julien Maury\/Flickr, Public Domain Mark 1.0<\/figcaption><\/figure>\n<\/div>\n
<\/div>\n

 <\/p>\n

These seemingly unrelated examples and many more highlight mathematical relationships that we associate with beauty in artistic form. These include the “Rule of Thirds” and ” the “Golden Ratio”.<\/p>\n

Golden Ratio<\/h3>\n

The golden ratio<\/strong>, also known as the golden proportion or the divine proportion, is a ratio aspect that can be found in beauty from nature to human anatomy as well as in golden rectangles that are commonly found in building structures.<\/p>\n

The golden ratio is expressed in nature from plants to creatures such as starfish, honeybees, seashells, and more. It is commonly noted by the Greek letter [latex]\\phi[\/latex], pronounced “fee”.<\/p>\n

\n

[latex]\\phi=\\frac{1+\\sqrt{5}}{2}[\/latex], which has a decimal value approximately equal to [latex]1.618[\/latex].<\/p>\n<\/section>\n

\n
\n

golden ratio<\/h3>\n

Often represented by the Greek letter Phi ([latex]\\phi[\/latex]), the Golden Ratio<\/strong> (approximately [latex]1:1.618[\/latex]) has been used in art and architecture to create aesthetically pleasing proportions.\u00a0<\/div>\n<\/section>\n

The golden ratio has been used by artists through the years and can be found in art dating back to 3000 BC. Leonardo da Vinci is considered one of the artists who mastered the mathematics of the golden ratio, which is prevalent in his artwork such as Virtuvian Man (seen below). This famous masterpiece highlights the golden ratio in the proportions of an ideal body shape.<\/p>\n

\n
\"Vitruvian
Figure 2. The Virtuvian Man by Leonardo da Vinci<\/figcaption><\/figure>\n<\/div>\n

 <\/p>\n

The golden ratio is approximated in several physical measurements of the human body and parts exhibiting the golden ratio are simply called golden. The ratio of a person\u2019s height to the length from their belly button to the floor is [latex]\\phi[\/latex] or approximately [latex]1.618[\/latex]. The bones in our fingers (excluding the thumb), are golden as they form a ratio that approximates [latex]\\phi[\/latex]. The human face also includes several ratios and those faces that are considered attractive commonly exhibit golden ratios.<\/p>\n

If a person\u2019s height is [latex]5[\/latex] ft [latex]6[\/latex] in, what is the approximate length from their belly button to the floor rounded to the nearest inch, assuming the ratio is golden? <\/p>\n