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This is the geometric relation between π and sqrt(2) In this picture, we can clearly see that the circle's radius is “1”. The side of the pink square is sqrt(2) because 1^2+1^2=c^2, c=sqrt(2). The purple arc on top of sqrt(2) is “π/2”. This picture is a way to calculate π. To Be Continued

Detailed information on the eyesight of butterflies: Butterflies carry proteins called cryptochrome that is sensitive to blue lights Butterflies can see magnetic fields, which is named magnetoception To butterflies, the sky is stripped because they see parabolic, which are the magnetic fields Most of the butterflies can see 4 colors. Some of them can see five or 5 and can see colors we can’t see Butterflies can see UV light, which can help them detect nectar Butterflies can perceive polarized light or light waves that move in only one direction “What Life Is like for a Butterfly.” Anupam Katkar, Anupam Katkar, 28 Jan. 2016, anupamkatkar.com/2016/01/28/butterfly-intelligence-vision-senses/. Accessed 27 Nov. 2022. Undetailed information on the senses of butterflies: touch larvae Touch through hairs that extend through sockets on the exoskeleton Tactile setae are scattered evenly over the whole body Have a variety of responses to touch, which may change over time adult Touch through hairs that extend through sockets on the exoskeleton Tactile setae are scattered evenly over the whole body Specialized setae and nerves can help them sense wind, gravity, and the position of the head, body, wings, legs, antennae, etc hearing larvae Perceive sound through tactile setae Mainly respond to sudden noises (habituation) adult Sense sound through veins in their wings sight larvae Limited and poor See the same range of light as adults See with 12 ocelli adult See with thousands of ommatidia See with compound eyes

The important video that you should watch before reading this article: https://www.youtube.com/watch?v=gB9n2gHsHN4&ab_channel=3Blue1Brown In this article, I will write about how I calculated the dimension of the coastline of Taiwan. I put an image of the Taiwan map with a scale factor of 1 into GeoGebra, and after that, I put 1*1 squares on the places the coastline had touched. As a result, there were 27 squares that the coastline had touched. In the end, it looked like Fig 1, and the number of squares that had touched the coastline in different scale factors is in Table 1. Fig 1 Table 1 According to the video I posted, the graph will be the log of the scaling factor on the x-axis and the log of the number of squares that had touched the coastline on the y-axis, like in Fig 3. So the result should be like in Table 2. Fig 2 Table 2 After drawing the prediction line of Fig Table out, I noticed something was wrong. According to the video I provided, the slope of the prediction line is the dimension of the picture, but the slope of the prediction line is about 0.89, while I think it will be over 1. Maybe I had counted wrong when the number was too big, or there wasn't enough data. Another possible reason is that the squares are not very precise. I should make them smaller. Fig 3

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