I looked at Howard models and found out that a saddle point existed when the model contained an hyperbolic region. But if no hyperbolic region is there - ...
I thought the "saddle point" existed when a hyperbolic surface is embedded in 3D Euclidean space. If the surface were embedded in higher dimensions, would it...
There should be saddle points in higher dimensions too according to the definition: a saddle point means a point which is a minimum in one plane and maximum in...
I am thinking that a saddle shaped surface in 4-space would be recognizable as saddle shaped, but that it would look really flat. Perhaps how we view the ...
I've been pretty swamped for a couple weeks, but I have been reading the postings with interest.<br><br>I have been looking at closed S-manifolds, and one that...
A way of categorizing Smarandache geometries has occurred to me. We can categorize according to how an axiom is denied.<br><br>On a basic level, there are two ...
<I've been pretty swamped for a couple weeks, but I have been reading the postings with interest.><br><br>We missed you, Howard, welcome back.<br>I...
Howard found 2-D Smarandache geometries. Can someone find 3-D or higher dimension such geometry?<br>I know than Ben Saucer was interested in such higher...
I printed Howard's 13-th picture and tried to assemble it. First I got a cylinder, then it became difficult to connect the same points, I mean to get a Klein ...
<<What about only elliptic region and hyperbolic region in your first model?<br>Isn't it negating S-denial?>><br><br>In that first model, every...
I'm pretty sure that this is the case. I'm not ready to work on this now, but someone else can. <br><br>Here's an idea to maybe start with. Start with R^3. The...
<<I printed Howard's 13-th picture and tried to assemble it. First I got a cylinder, then it became difficult to connect the same points, I mean to get a...
<<Here's an idea to maybe start with. Start with R^3. The coordinate planes divide this into octants. Cut the space along the xz-plane where x is ...
<<<I printed Howard's 13-th picture and tried to assemble it. First I got a<br> cylinder, then it became difficult to connect the same points, I mean...
What idea!<br>What about enveloping any solid with triangles, of course forcing it, I mean folded triangles to be able to approximatively enveloping the ...
You mean a model constructed by gluing simplexes together? In 2D, six triangles meet at the corners to make a Euclidean point. Five or less make a pyramid ...
How to construct a Klein bottle in 3D Euclidean space: Begin with a flat horizontal plane. Cut two holes side by side. On the top side of the plane, attach a ...
Klein bottles are known to be "Euclidean". But when we are dealing with compact manifolds, rather than infinite planes, it IS possible for lines to intersect...
<<In a compact manifold,<br> you cannot always "extend a line to ANY length", or "produce a UNIQUE<br> line through two points", or "draw a circle having...
<<In 3D, use regular tetrahedra. Five of them meeting an edge makes slightly<br> LESS than a Euclidean segment, but not quite. It makes a 3D elliptic<br>...
<<I come back to say that by Smarandache manifold I understand a manifold that supports a Smarandache geometry. >><br><br>What I'm calling a ...
<<How to construct a Klein bottle in 3D Euclidean space>><br><br>This is a great idea! I think you can attach the ends of the tubes at infinity two...
<<Can I have some clear definitions of an "open manifold", "close manifold", and of a "compact manifold"?>><br><br>Any manifold that is embedded in...
<<<I come back to say that by Smarandache manifold I understand a manifold that supports a Smarandache geometry. >><br><br>What I'm calling a ...
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