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      "body": "![Axe of Al-Hazan 2 blades corrected.jpg](https://cdn.steemitimages.com/DQmSj4xpBy1z6aFqLi2xXgq4nQAkoBnVX4xyMvW9AsHejsf/Axe%20of%20Al-Hazan%202%20blades%20corrected.jpg)\n\nAbove, you see the axe of Al-Hazan. It consists of two blades and the handle. We don't care about the handle. Each blade is bounded by two small quarter circle arcs and one larger one. The center of the larger quarter arcs is at point O. The distance from O to the edge of the blades is 1 unit. Your task is to find out how big the blades are. Specifically, what is the area of the red shaded regions?\n\nTake a crack at solving the puzzle before scrolling down for my solution. There are several ways to approach the problem. I suspect mine is the laziest.\n\nAl-Hazan in the title is an unfortunate misspelling of the name of an Arab mathematician and physicist, Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham al-Basri. al-Basri had some famous lunes shown below, but, to the best of my knowledge, no axe.\n![lunes of alhazen.jpg](https://cdn.steemitimages.com/DQmaCNUphw8a7P7qGfQmmNdBY6d5idfKG5XDFgNyQiMjoLP/lunes%20of%20alhazen.jpg)\nMy solution:\n\n![Axe of Al-Hazan with aux lines 2 blades.jpg](https://cdn.steemitimages.com/DQmPQGwi68iHS7HGbRuUPrZBNtrZKSPff1wBfme5X1LHjdM/Axe%20of%20Al-Hazan%20with%20aux%20lines%202%20blades.jpg)\n\nAdd 3 auxiliary lines as shown. Because the edge of the blade is a quarter circle, the central angle at point O is a right angle and the triangle is a right triangle. \n![Axe of Al-Hazan with labels 2 blades.jpg](https://cdn.steemitimages.com/DQmaTQjtNaJQNVRbToa86dDzTdTUaKC99N8Tf16EP4QnP53/Axe%20of%20Al-Hazan%20with%20labels%202%20blades.jpg)\n\nThe areas labelled A, B and C are circle segments bounded by a quarter circle and the associated chord. They are therefore similar. That is, they have the same shape.\n![Axe of Al-Hazan pyth 2 blades.jpg](https://cdn.steemitimages.com/DQmZYb9J9qWuSUzVqBBV4tbeVkKJBEC5QguiafXGQfSFDpk/Axe%20of%20Al-Hazan%20pyth%202%20blades.jpg)\n\nA, B and C are similar shapes constructed on the sides and hypotenuse of a right triangle, so the area of A + area of B = area of C. The blade consists of C plus the red wedge shape inside the triangle. The triangle consists of the red wedge, plus A, plus B. So the area of the blade is equal to the area of the triangle. The triangle has sides of length 1, the distance from the center of the circle to the edge of the blade. Therefore the triangle's area is (1 x 1)/2 = 1/2. There are 2 blades, so their combined area is 1.",
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      "voter": "stive",
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    {
      "author": "stive",
      "body": "Very nice. Will you be following up with a post on arithmetic with Zeckendorf notation?",
      "json_metadata": "{\"tags\":[\"mathematics\"],\"app\":\"steemit/0.1\"}",
      "parent_author": "nenio",
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{
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    {
      "author": "stive",
      "body": "\"And if you're the 23rd reader, there is a 50% possibility that it is your bday, so happy birthday :)\"\n\nOops! And you were doing so well. The probability that today (or rather yesterday when you posted) is the 23rd reader's birthday is 1/365, less than 3%.  The probability that it is the birthday of one of the first 23 readers is still only 1 - (364/365)^23 , less than 7%. \n\nAs you showed above, it is only when each pair of birthdays are compared that the probability of a match reaches 50% for 23 people.",
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      "author": "leaky20",
      "body": "Wow. nice analysis.  I agree with a lot of what you are saying. You might be right that certain characteristics (increased respiratory systems) wont offer a reproductive advantage. But evolution works in many ways - not just natural selection. Over long periods of time who knows what would happen.\n\nI agree that plants will mostly be selectively bred. However, I do think that plants and animals (if animals are brought to Mars) will follow a natural evolutionary path as well. For instance, bacteria and pathogens will exist and they will evolve naturally. As such, humans, animals, and plants will all have to adapt to the changing threat of pathogens. So though I certainly agree that plant evolution will mostly be influenced by humans, I still think that there will be a natural component as well. Seeds, algae's and things like that would likely escape into the atmosphere and spread throughout the planet over time.\n\nYou sound pretty knowledgeable on the subject and gave me a lot to think about.  Thanks for the comment! I'm interested in hearing more of your thoughts.",
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    {
      "author": "zenithwombat",
      "body": "@@ -571,18 +571,33 @@\n hat \n-why\n  would \n+be the purpose of \n huma\n@@ -619,77 +619,237 @@\n ving\n- the planet have any purpose? what would be the financial incentive? \n+? What would be the financial incentive? Over population issues seem as if they could become moot as well, as we would all be residing within a digital network... perhaps one that eventually would span the galaxy? (crazy laugh)%0A%0A\n %0A%0A%0Ah\n",
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{
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    {
      "author": "zenithwombat",
      "body": "@@ -392,16 +392,296 @@\n evant.%0A%0A\n+So if this is the case, and we advance VR technology to the point where we can experience an environment remotely via a robotic explorer with instantaneous transmission, what why would humans actually leaving the planet have any purpose? what would be the financial incentive? %0A%0A%0A\n https://\n",
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      "author": "zenithwombat",
      "body": "@@ -390,8 +390,146 @@\n elevant.\n+%0A%0Ahttps://www.picoquant.com/applications/category/quantum-optics/quantum-communication%0A%0Ahttps://en.wikipedia.org/wiki/Quantum_entanglement\n",
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      "author": "zenithwombat",
      "body": "A new field in communication is arising in the form of quantum communication.  I don't know much about it but at this point the focus seems to be on cryptography. However if we can use quantum entanglement in communication it seems to me that eventually we will be able to send communication or transmissions over any distance instantaneously.  This would make the speed of light limit irrelevant.",
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      "author": "steemitboard",
      "body": "Congratulations @stive! You have completed some achievement on Steemit and have been rewarded with new badge(s) :\n\n[![](https://steemitimages.com/70x80/http://steemitboard.com/notifications/firstvote.png)](http://steemitboard.com/@stive) You made your First Vote\n[![](https://steemitimages.com/70x80/http://steemitboard.com/notifications/firstpost.png)](http://steemitboard.com/@stive) You published your First Post\n[![](https://steemitimages.com/70x80/http://steemitboard.com/notifications/firstcomment.png)](http://steemitboard.com/@stive) You made your First Comment\n[![](https://steemitimages.com/70x80/http://steemitboard.com/notifications/firstvoted.png)](http://steemitboard.com/@stive) You got a First Vote\n\nClick on any badge to view your own Board of Honor on SteemitBoard.\nFor more information about SteemitBoard, click [here](https://steemit.com/@steemitboard)\n\nIf you no longer want to receive notifications, reply to this comment with the word `STOP`\n\n> By upvoting this notification, you can help all Steemit users. Learn how [here](https://steemit.com/steemitboard/@steemitboard/http-i-cubeupload-com-7ciqeo-png)!",
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      "author": "stive",
      "body": "Edwin Aldrin's nickname was Buzz. Buzz Lightyear from Toy Story was named after him.",
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      "author": "stive",
      "body": "The post is just a bad translation of material cut and pasted from another source. The cut and pasting could be excused if value, in the form of a competent translation had been added. As it is, most of the post is gibberish.",
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      "author": "stive",
      "body": "To the best of my knowledge, the current atmosphere of Mars, to the extent it has one at all, cannot sustain us. Presumably then you are presuming some sort of terraforming. If the atmosphere is terraformed, there is no reason to believe that it will differ in chemical makeup from that of Earth. The big difference will be density due to the lower gravity. People with larger lungs or a more efficient respiratory system would have an advantage. But is it not at all clear that being able to labor more effectively in the field will give someone a reproductive advantage. Less advantaged individuals working in an office environment could compensate with a breathing aid.\n\nThe varieties of plants that humans eat do not evolve by natural selection. They evolve by artificial selection. That artificial selection is more driven by culture than by dietary needs. For example, maroon carrots have more beta carotene and are sweeter than orange carrots, but people think they're weird so they've never caught on. I suspect humans on Mars will develop strains of terrestrial plants that taste and look as much like the ones they left behind on Earth as agricultural science can achieve.",
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      "author": "stive",
      "body": "So many questions, so little time. Real life, in the form of my cats, is insisting that I pay attention to it, but I want to address some of the ideas you have raised.\n\nFirst, unless physicists are very wrong about the speed of light being a limitation, we won't be able to effectively operate machines at great distances. Response time lags of several years will make that impossible. We will not be able to explore the universe from the comfort of home.\n\nA more likely solution to the frailty of human bodies would be human-machine hybrids. At their most extreme, such hybrids would consist of a brain controlling a mechanical body designed to sustain the brain and to protect it from the thousand natural shocks (and ionizing radiation) that flesh is heir to.",
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      "author": "rebeccabe",
      "body": "welcome get to it",
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      "author": "saluton",
      "body": "Hello!",
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