Exploring Mathematical Approximations and Artistic Ratios
Uncover the world of mathematical approximations and artistic ratios through images and explanations, including the Vitruvian Man, the Golden Rectangle, and more. Dive into mathematical concepts and their visual representation, alongside insights into order approximations and first-order approximations.
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Presentation Transcript
The Vitruvian man http://blog.world-mysteries.com/science/vitruvian-man-by- leonardo-da-vinci/
Golden rectangle ?+? ? = ? ?= ? By Ahecht (Original); Pbroks13 (Derivative work); Joo. (Editing) - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=15656641
The Vitruvian man NOT IN GOLDEN RATIO http://blog.world-mysteries.com/science/vitruvian-man-by- leonardo-da-vinci/
The Sacrament of Last Supper by Salvador Dali IN GOLDEN RATIO By National Gallery of Art, Washington D.C., Fair use, https://en.wikipedia.org/w/index.php?curid=24679655
Approximations What is sin 0 ? What is log 1 ? What is arctan 0 ? What is sin 0.25 ? What is log 1.3 ? What is arctan 1.1 ? Graphed using desmos.com We want NUMBERS! (Or something close enough)
Approximations What is sin 0 = 0 What is log 1 = 0 What is arctan 0 = 0 What is sin 0.25 ? What is log 1.3 ? What is arctan 1.1 ? We want NUMBERS! (Or something close enough)
Zeroeth Order Approximation The approximating function would have order zero (aka constant) Given a function ?(?) on the real line Fix a point ? Want a function ?(?)to approximate the value of ?(?) near (and at) ?. We define a function ? ? = ?(?).
Example 1 ? ? = ?? ? = 0 ? ? = ?0= 1 Graphed using desmos.com ? 0.1 = 1 ,?0.1= 1.10517091808 At ? = 0.1,
Remark This would only work if we knew what ? ? was ? Can t do this for ? ? = ?? and ? = 14 coz we don t know what ?14is.
First order approximation Approximating function ? ? is of first order ? ? = ? + ?? Given a (differentiable) function ?(?) on the real line Fix a point ? Choose ? = ? ? ? ? ? and ? = ? ? ? ? = ? ? ? ? ? + ? ? ? ? ? = ? ? + ? (?)(? ?)
Example 2 ? ? = ?? ? = 0 ? ? = ?0+ 1 ? 0 ? ? = 1 + ? At ? = 0.1, ? 0.1 = 1.1 ?0.1= 1.10517091808
Example 3 Find an approximate value of 5 ? Identify a function ?(?) Make an (intelligent) choice of a point ? Find ? ?
Example 3 Find an approximate value of 5 ? Identify a function ?(?) Make an (intelligent) choice of a point ? Find ? ? ? ? = ? = 4 (good choice as we know ? ? = ? ? = 2 + (? 4)1 ? 4 = 2) 4 5 4 4 ? 5 = 2 + = 2 + 0.25 = 2.25 5 = 2.236
Example 4 Approximate ? ? (Hint: this function came before)
? ? = arctan(?) ? = 0
Finding the Golden ratio! (drum roll) ?+? ? = ? ?= ? ?=? ? ? ?+? 1 +1 ?= ? ? + 1 = ?2 ?2 ? 1 = 0 ? is a solution of the polynomial ? ? = ?2 ? 1 By Max Ronnersj - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=19108050
? =1+ 5 1+2.25 = 1.625 2 2 ? = 1.618033
Second order approximation Approximating function ? ? is of second order (degree 2) ? ? = ? + ?? + ??2 Given a (differentiable) function ?(?) on the real line Fix a point ? ? ? = ? ? + ? ? ? ? +? ? ? ?2 2
Example 5 Approximate ?0.1 using second order approximation?
Example 6 Approximate 5 using second order approximation?
To sum it all up ? ? = ?? Zeroeth order approximation ? ? = ? ? First order approximation ? ? = ? ? + ? (?)(? ?) Second order approximation ? ? = ? ? + ? ? ? ? +? ? ? ?2 2
Higher order approximation aka Taylor polynomials 3?? order approximation ? ? = ? ? + ? ? ? ? +? ? ? ?2+? ? ? ?3 2 3! ?? order approximation ? ? = ? ? + ? ? ? ? +? ? ? ?2+ +?(?)? ? ?? 2 ?! ??(?)? ? ?? ? ? = 0 ?!
Compute the 5? order Taylor polynomial for log ? about ? = 1.
Compute the 5? order Taylor polynomial for log ? about ? = 1. And the final slide!
