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      "body": "<html>\n<p><img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Dimension_levels.svg/385px-Dimension_levels.svg.png\" width=\"385\" height=\"129\"/></p>\n<p><br></p>\n<p>It is not easy for most of us to think of high dimensions because it is hard to visualize something more than three dimensions. Even visualizing three dimensions is not easy sometimes. However, we are curious when we heard scientists talk about 4-D, 5-D or even 11-D, how do they look like?</p>\n<p><br>\nAlthough there are lots of ways to visualize high dimensions with tools of dimensionality reduction, it is difficult for us to get the intuition of 4-D, 5-D inside our head. Therefore I would like to share how I view them in my head.<br>\nIn physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the number of coordinates needed to specify a point on the object. For example, you can specify a point on a number line with one number, you can count them -5, -4, -3 and so on.</p>\n<p><img src=\"http://img.sparknotes.com/figures/5/50ca5e784bb7e4242910d5b8a571d103/number_line.gif\" width=\"438\" height=\"152\"/></p>\n<p>Fig.1.</p>\n<p><br>\nIn two dimensions, 2 numbers are needed to represent a point.</p>\n<p><img src=\"https://mathbitsnotebook.com/Geometry/Transformations/refX1.jpg\" width=\"300\" height=\"300\"/></p>\n<p>Fig.2.<br>\n</p>\n<p>If you want to specify point A, it would be (-2,1).</p>\n<p><br>\nIn three dimensions, 3 numbers are needed to specify a point.</p>\n<p><img src=\"https://i.ytimg.com/vi/VA5AmjhTA3A/maxresdefault.jpg\" width=\"1920\" height=\"1086\"/></p>\n<p>Fig.3.</p>\n<p><br>\nYou can see the blue point is specified as (3,4,5) on the three dimensionals plane.</p>\n<p><br>\nAs you can guess, a point needed to be specified by 4 numbers in a four dimensional plane. Let say a point’s coordination is (3,4,5,6). Where 6 is its coordinate on the forth dimension. Then, you can imagine a very long number line like the one dimension. After that, imagine there is a three dimensional world lies on each point on the number line. You can think it as a chain of 3-D box (like Fig.3.) , from box -3, box -2, box -1 &nbsp;…</p>\n<p><img src=\"https://holland.pk/7u3r1jte\" width=\"800\" height=\"559\"/><br>\nThe point (3,4,5,6) is then on the (3,4,5) of box 6.You can also think of the forth dimension as “Time” as an analogy. The four dimensional world is like a chain of video frames.</p>\n<p><img src=\"https://www.engineersgarage.com/sites/default/files/imagecache/Original/wysiwyg_imageupload/4214/How-Video-is-displayed2.jpg\" width=\"599\" height=\"397\"/><br>\n<br>\nThe (3,4,5,6) point is representing the point (3,4,5) on the sixth time frame!</p>\n<p><br>\nIt is similar for higher dimensions. For example, a 5-D will be a number line, with each of the point having 4-D world. If you think of 4-D as time. Then 5-D will be parallel universe. As every possible position of particles are described in the 3-D world, the past and future position of particles are described in the 4-D. Then it should be a new story of universe in each coordinates of the fifth dimension.</p>\n<p><br>\nIn short, practically in handling high dimensional data, data scientists will not think of the data in this way. But isn’t it interesting to think of high dimensions in this way? Does it give you some inspiration on how the world works?<br>\n</p>\n<p>Image Source <a href=\"https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Dimension_levels.svg/385px-Dimension_levels.svg.png\">1</a>-<a href=\"http://img.sparknotes.com/figures/5/50ca5e784bb7e4242910d5b8a571d103/number_line.gif\">2</a>-<a href=\"https://mathbitsnotebook.com/Geometry/Transformations/refX1.jpg\">3</a>-<a href=\"https://i.ytimg.com/vi/VA5AmjhTA3A/maxresdefault.jpg\">4</a>-<a href=\"https://www.engineersgarage.com/sites/default/files/imagecache/Original/wysiwyg_imageupload/4214/How-Video-is-displayed2.jpg\">5</a></p>\n</html>",
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      "body": "<html>\n<p><img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Dimension_levels.svg/385px-Dimension_levels.svg.png\" width=\"385\" height=\"129\"/></p>\n<p><br></p>\n<p>It is not easy for most of us to think of high dimensions because it is hard to visualize something more than three dimensions. Even visualizing three dimensions is not easy sometimes. However, we are curious when we heard scientists talk about 4-D, 5-D or even 11-D, how do they look like?</p>\n<p><br>\nAlthough there are lots of ways to visualize high dimensions with tools of dimensionality reduction, it is difficult for us to get the intuition of 4-D, 5-D inside our head. Therefore I would like to share how I view them in my head.<br>\nIn physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the number of coordinates needed to specify a point on the object. For example, you can specify a point on a number line with one number, you can count them -5, -4, -3 and so on.</p>\n<p><img src=\"http://img.sparknotes.com/figures/5/50ca5e784bb7e4242910d5b8a571d103/number_line.gif\" width=\"438\" height=\"152\"/></p>\n<p>Fig.1.</p>\n<p><br>\nIn two dimensions, 2 numbers are needed to represent a point.</p>\n<p><img src=\"https://mathbitsnotebook.com/Geometry/Transformations/refX1.jpg\" width=\"300\" height=\"300\"/></p>\n<p>Fig.2.<br>\n</p>\n<p>If you want to specify point A, it would be (-2,1).</p>\n<p><br>\nIn three dimensions, 3 numbers are needed to specify a point.</p>\n<p><img src=\"https://i.ytimg.com/vi/VA5AmjhTA3A/maxresdefault.jpg\" width=\"1920\" height=\"1086\"/></p>\n<p>Fig.3.</p>\n<p><br>\nYou can see the blue point is specified as (3,4,5) on the three dimensionals plane.</p>\n<p><br>\nAs you can guess, a point needed to be specified by 4 numbers in a four dimensionals plane. Let say a point’s coordination is (3,4,5,6). Where 6 is its coordinate on the forth dimension. Then, you can imagine a very long number line like the one dimension. After that, imagine there is a three dimensional world lies on each point on the number line. You can think it as a chain of 3-D box (like Fig.3.) , from box -3, box -2, box -1 &nbsp;…<br>\nThe point (3,4,5,6) is then on the (3,4,5) of box 6.You can also think of the forth dimension as “Time” as an analogy. The four dimensional worlds are like a chain of video frames.</p>\n<p><img src=\"https://www.engineersgarage.com/sites/default/files/imagecache/Original/wysiwyg_imageupload/4214/How-Video-is-displayed2.jpg\" width=\"599\" height=\"397\"/><br>\n<br>\nThe (3,4,5,6) point is representing the point (3,4,5) on the sixth time frame!</p>\n<p><br>\nIt is similar for higher dimensions. For example, a 5-D will be a number line, with each of the point having 4-D world. If you think of 4-D as time. Then 5-D will be parallel universe. As every possible position of particles are described in the 3-D world, the past and future position of particles are described in the 4-D. Then it should be a new story of universe in each coordinates of the fifth dimension.</p>\n<p><br>\nIn short, practically in handling high dimensional data, data scientists will not think of the data in this way. But isn’t it interesting to think of high dimensions in this way? Does it give you some inspiration on how the world works?<br>\n</p>\n<p>Image Source <a href=\"https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Dimension_levels.svg/385px-Dimension_levels.svg.png\">1</a>-<a href=\"http://img.sparknotes.com/figures/5/50ca5e784bb7e4242910d5b8a571d103/number_line.gif\">2</a>-<a href=\"https://mathbitsnotebook.com/Geometry/Transformations/refX1.jpg\">3</a>-<a href=\"https://i.ytimg.com/vi/VA5AmjhTA3A/maxresdefault.jpg\">4</a>-<a href=\"https://www.engineersgarage.com/sites/default/files/imagecache/Original/wysiwyg_imageupload/4214/How-Video-is-displayed2.jpg\">5</a></p>\n</html>",
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      "author": "mobbs",
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      "body": "@@ -444,18 +444,16 @@\n , how do\n-es\n  they lo\n@@ -512,18 +512,18 @@\n  to visu\n-l\n a\n+l\n ize high\n",
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      "body": "<html>\n<p><img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Dimension_levels.svg/385px-Dimension_levels.svg.png\" width=\"385\" height=\"129\"/></p>\n<p><br></p>\n<p>It is not easy for most of us to think of high dimensions because it is hard to visualize something more than three dimensions. Even visualizing three dimensions is not easy sometimes. However, we are curious when we heard scientists talk about 4-D, 5-D or even 11-D, how does they look like?</p>\n<p><br>\nAlthough there are lots of ways to visulaize high dimensions with tools of dimensionality reduction, it is difficult for us to get the intuition of 4-D, 5-D inside our head. Therefore I would like to share my how I view them in my head.<br>\nIn physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the number of coordinates needed to specify a point on the object. For example, you can specify a point on a number line with one number, you can count them -5, -4, -3 and so on.</p>\n<p><img src=\"http://img.sparknotes.com/figures/5/50ca5e784bb7e4242910d5b8a571d103/number_line.gif\" width=\"438\" height=\"152\"/></p>\n<p>Fig.1.</p>\n<p><br>\nIn two dimensions, 2 numbers are needed to represent a point.</p>\n<p><img src=\"https://mathbitsnotebook.com/Geometry/Transformations/refX1.jpg\" width=\"300\" height=\"300\"/></p>\n<p>Fig.2.<br>\n</p>\n<p>If you want to specify point A, it would be (-2,1).</p>\n<p><br>\nIn three dimensions, 3 numbers are needed to specify a point.</p>\n<p><img src=\"https://i.ytimg.com/vi/VA5AmjhTA3A/maxresdefault.jpg\" width=\"1920\" height=\"1086\"/></p>\n<p>Fig.3.</p>\n<p><br>\nYou can see the blue point is specified as (3,4,5) on the three dimensionals plane.</p>\n<p><br>\nAs you can guess, a point needed to be specified by 4 numbers in a four dimensionals plane. Let say a point’s coordination is (3,4,5,6). Where 6 is its coordinate on the forth dimension. Then, you can imagine a very long number line like the one dimension. After that, imagine there is a three dimensional world lies on each point on the number line. You can think it as a chain of 3-D box (like Fig.3.) , from box -3, box -2, box -1 &nbsp;…<br>\nThe point (3,4,5,6) is then on the (3,4,5) of box 6.You can also think of the forth dimension as “Time” as an analogy. The four dimensional worlds are like a chain of video frames.</p>\n<p><img src=\"https://www.engineersgarage.com/sites/default/files/imagecache/Original/wysiwyg_imageupload/4214/How-Video-is-displayed2.jpg\" width=\"599\" height=\"397\"/><br>\n<br>\nThe (3,4,5,6) point is representing the point (3,4,5) on the sixth time frame!</p>\n<p><br>\nIt is similar for higher dimensions. For example, a 5-D will be a number line, with each of the point having 4-D world. If you think of 4-D as time. Then 5-D will be parallel universe. As every possible position of particles are described in the 3-D world, the past and future position of particles are described in the 4-D. Then it should be a new story of universe in each coordinates of the fifth dimension.</p>\n<p><br>\nIn short, practically in handling high dimensional data, data scientists will not think of the data in this way. But isn’t it interesting to think of high dimensions in this way? Does it give you some inspiration on how the world works?<br>\n</p>\n<p>Image Source <a href=\"https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Dimension_levels.svg/385px-Dimension_levels.svg.png\">1</a>-<a href=\"http://img.sparknotes.com/figures/5/50ca5e784bb7e4242910d5b8a571d103/number_line.gif\">2</a>-<a href=\"https://mathbitsnotebook.com/Geometry/Transformations/refX1.jpg\">3</a>-<a href=\"https://i.ytimg.com/vi/VA5AmjhTA3A/maxresdefault.jpg\">4</a>-<a href=\"https://www.engineersgarage.com/sites/default/files/imagecache/Original/wysiwyg_imageupload/4214/How-Video-is-displayed2.jpg\">5</a></p>\n</html>",
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      "body": "Thank you!",
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      "body": "<html>\n<p><img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Dimension_levels.svg/385px-Dimension_levels.svg.png\" width=\"385\" height=\"129\"/></p>\n<p><br></p>\n<p>It is not easy for most of us to think of high dimensions because it is hard to visualize something more than three dimensions. Even visualizing three dimensions is not easy sometimes. However, we are curious when we heard scientists talk about 4-D, 5-D or even 11-D, how does they look like?</p>\n<p><br>\nAlthough there are lots of ways to visulaize high dimensions with tools of dimensionality reduction, it is difficult for us to get the intuition of 4-D, 5-D inside our head. Therefore I would like to share my how I view them in my head.<br>\nIn physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the number of coordinates needed to specify a point on the object. For example, you can specify a point on a number line with one number, you can count them -5, -4, -3 and so on.</p>\n<p><img src=\"http://img.sparknotes.com/figures/5/50ca5e784bb7e4242910d5b8a571d103/number_line.gif\" width=\"438\" height=\"152\"/></p>\n<p>Fig.1.</p>\n<p><br>\nIn two dimensions, 2 numbers are needed to represent a point.</p>\n<p><img src=\"http://www.mathwarehouse.com/transformations/images/reflections/picture-in-x-axis.gif\"/><img src=\"http://www.mathopenref.com/images/coordinates/coords.png\"/><img src=\"https://mathbitsnotebook.com/Geometry/Transformations/refX1.jpg\" width=\"300\" height=\"300\"/></p>\n<p>Fig.2.<br>\n</p>\n<p>If you want to specify point A, it would be (-2,1).</p>\n<p><br>\nIn three dimensions, 3 numbers are needed to specify a point.</p>\n<p><img src=\"https://i.ytimg.com/vi/VA5AmjhTA3A/maxresdefault.jpg\" width=\"1920\" height=\"1086\"/></p>\n<p>Fig.3.</p>\n<p><br>\nYou can see the blue point is specified as (3,4,5) on the three dimensionals plane.</p>\n<p><br>\nAs you can guess, a point needed to be specified by 4 numbers in a four dimensionals plane. Let say a point’s coordination is (3,4,5,6). Where 6 is its coordinate on the forth dimension. Then, you can imagine a very long number line like the one dimension. After that, imagine there is a three dimensional world lies on each point on the number line. You can think it as a chain of 3-D box (like Fig.3.) , from box -3, box -2, box -1 &nbsp;…<br>\nThe point (3,4,5,6) is then on the (3,4,5) of box 6.You can also think of the forth dimension as “Time” as an analogy. The four dimensional worlds are like a chain of video frames.</p>\n<p><img src=\"https://www.engineersgarage.com/sites/default/files/imagecache/Original/wysiwyg_imageupload/4214/How-Video-is-displayed2.jpg\" width=\"599\" height=\"397\"/><br>\n<br>\nThe (3,4,5,6) point is representing the point (3,4,5) on the sixth time frame!</p>\n<p><br>\nIt is similar for higher dimensions. For example, a 5-D will be a number line, with each of the point having 4-D world. If you think of 4-D as time. Then 5-D will be parallel universe. As every possible position of particles are described in the 3-D world, the past and future position of particles are described in the 4-D. Then it should be a new story of universe in each coordinates of the fifth dimension.</p>\n<p><br>\nIn short, practically in handling high dimensional data, data scientists will not think of the data in this way. But isn’t it interesting to think of high dimensions in this way? Does it give you some inspiration on how the world works?<br>\n</p>\n<p>Image Source <a href=\"https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Dimension_levels.svg/385px-Dimension_levels.svg.png\">1</a>-<a href=\"http://img.sparknotes.com/figures/5/50ca5e784bb7e4242910d5b8a571d103/number_line.gif\">2</a>-<a href=\"https://mathbitsnotebook.com/Geometry/Transformations/refX1.jpg\">3</a>-<a href=\"https://i.ytimg.com/vi/VA5AmjhTA3A/maxresdefault.jpg\">4</a>-<a href=\"https://www.engineersgarage.com/sites/default/files/imagecache/Original/wysiwyg_imageupload/4214/How-Video-is-displayed2.jpg\">5</a></p>\n</html>",
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