<story><title>Spikey Spheres</title><url>http://www.solipsys.co.uk/new/SpikeySpheres.html?HN_20161120</url></story><parent_chain><item><author>jakobegger</author><text>A teacher of mine told us, &amp;quot;Don&amp;#x27;t try to imagine 4D space. You can&amp;#x27;t. If you think you can imagine a 4D hypersphere, you&amp;#x27;re wrong. It&amp;#x27;s just not possible.&amp;quot;&lt;p&gt;And everytime someone talks about how to imagine higher-dimensional space, I have to think back, and I&amp;#x27;m convinced that my teacher was right. Our intuitive understanding of 2D and 3D just doesn&amp;#x27;t generalise to higher dimensions.&lt;p&gt;A hypersphere is nothing like a sphere, just like a sphere is nothing like a circle, which is again nothing like a line segment.&lt;p&gt;If you talk about multidimensional geometries, stick to the math, and don&amp;#x27;t try imagining it. Analogies to 2D or 3D objects (it&amp;#x27;s smooth, but like a spike) are pointless and don&amp;#x27;t lead to new insights. We really need to stick to precise, mathematical language if we want to work with higher dimensional geometry.</text></item></parent_chain><comment><author>jobigoud</author><text>I think he was wrong. It&amp;#x27;s pretty hard but not impossible. You don&amp;#x27;t visualize them directly as 3D objects though, you visualize 3D objects with metadata attached everywhere. While you navigate the space in your mind&amp;#x27;s eye you have an extra layer of information that flows with you.&lt;p&gt;Consider Alicia Boole Stott. She created cardboard models of the various cross-sections of the 6 regular polytopes. Including the 24-cell which has no equivalent in 3D, the 120-cell and the 600-cell which are super complex figures. I think if you read her story you&amp;#x27;ll also be convinced that she indeed visualized the figures. [1]&lt;p&gt;I spent quite a lot of time earlier this year on this visualization exercise on the 4 simplest regular polytopes while working on an toy application to manipulate 4D objects and tesseractic honeycomb in VR.&lt;p&gt;You can have a good grasp of where every edge go and the angle between every faces. I could count the number of vertices and edges of a 16-cell or 24-cell just by navigating around the figure in my head. The tesseract and 16-cell are good starting points.&lt;p&gt;1: &lt;a href=&quot;http:&amp;#x2F;&amp;#x2F;www.sciencedirect.com&amp;#x2F;science&amp;#x2F;article&amp;#x2F;pii&amp;#x2F;S0315086007000973&quot; rel=&quot;nofollow&quot;&gt;http:&amp;#x2F;&amp;#x2F;www.sciencedirect.com&amp;#x2F;science&amp;#x2F;article&amp;#x2F;pii&amp;#x2F;S0315086007...&lt;/a&gt;</text></comment>
<story><title>Spikey Spheres</title><url>http://www.solipsys.co.uk/new/SpikeySpheres.html?HN_20161120</url></story><parent_chain><item><author>jakobegger</author><text>A teacher of mine told us, &amp;quot;Don&amp;#x27;t try to imagine 4D space. You can&amp;#x27;t. If you think you can imagine a 4D hypersphere, you&amp;#x27;re wrong. It&amp;#x27;s just not possible.&amp;quot;&lt;p&gt;And everytime someone talks about how to imagine higher-dimensional space, I have to think back, and I&amp;#x27;m convinced that my teacher was right. Our intuitive understanding of 2D and 3D just doesn&amp;#x27;t generalise to higher dimensions.&lt;p&gt;A hypersphere is nothing like a sphere, just like a sphere is nothing like a circle, which is again nothing like a line segment.&lt;p&gt;If you talk about multidimensional geometries, stick to the math, and don&amp;#x27;t try imagining it. Analogies to 2D or 3D objects (it&amp;#x27;s smooth, but like a spike) are pointless and don&amp;#x27;t lead to new insights. We really need to stick to precise, mathematical language if we want to work with higher dimensional geometry.</text></item></parent_chain><comment><author>jacobolus</author><text>However there is a 4-dimensional sphere (3-sphere) that you can easily access: the possible rotations of any object in 3-dimensional space is topologically and metrically isomorphic to a (half of a) 3-sphere, in the same way as the rotations of an object in 2-dimensional space are isomorphic to a 1-sphere (circle).&lt;p&gt;Think of an object’s current orientation as the “north pole” of our abstract 4-dimensional sphere. Rotating the object around any axis pushes you toward the equator, which you reach once you have rotated the object halfway around (the equator is made from all 180° turns, and is topologically a 2-sphere). Continuing to rotate the object takes you towards the “south pole”, which is where you get if you rotate your object 360°. To get back to the north pole, rotate by 360° again.&lt;p&gt;Cf. &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Versor&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Versor&lt;/a&gt; &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Quaternions_and_spatial_rotation&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Quaternions_and_spatial_rotati...&lt;/a&gt; and &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Rotation_group_SO(3)&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Rotation_group_SO(3)&lt;/a&gt;&lt;p&gt;* * *&lt;p&gt;One other way you can get at the 3-sphere is by taking a stereographic projection into all of 3-dimensional space (plus a point at infinity). If you have some 3-dimensional shapes drawn on the surface of your 3-sphere, these will get distorted under the stereographic projection, but locally angles will be preserved, just like a stereographic projection of the 2-sphere onto a piece of paper.</text></comment>