Janus

Have both parties shifted over time? Yes. But to put in in terms of one political analyst, While the Democrats may have shifted from their 45 yard line to their 35 yard line, the Republicans have shifted from their 45 yard line to somewhere behind their own goal post.

Agreed. In fact, I find it amazing that someone could be so wrong about so many different things in just one article!

Cutting the segments up while remaining on the sphere is no problem, you get this: However, going past this step becomes problematic. Each division is now an irregular triangle and won't divide into three similar triangles.,

It can't be done by dividing each section into diamonds,and have all the divisions have identical shapes. Look at the sphere on the left in post #27. Look at where the orange, blue and red sections meet, the interior angles of each at that point is 120 degrees. Now look at the left point of the orange section. It meets with a blue, red, and yellow section at this point and the interior angles are 90 degrees. If you draw lines along the surface from the midpoint of each side of the section to the midpoint of the other side, and then do the same for the other two sides, they cross in the middle at right angles to each other and are at a right angle were they meet the sides. You have divided the section into four pieces. Call them Starting at the top and going clockwise, call them North, East, South and West. The East and West pieces both have 90 degree interior angle at each corner. However, the North and South pieces have 3 90 degree corners and 1 120 degree corner. Not only that but the four corners of the curved section with the 120 degree interior angle don't lie in the same plane, so you can't even flat surface for a polygon side out of it. Changing how you slice up the section doesn't help if you keep the existing interior angles intact. Each piece would have to have at least 1 90 degree and 1 120 degree interior angle, and there is no way to get four identical pieces with this requirement to fit the existing section. I know that it looks like it should be possible at first glance, but you have to take into consideration that you are working with a curved surface. You can take another route, which is divide the section up from the corners. Draw a line from each corner to the opposite corner. You have now divided the section into 4 triangular pieces that each have a 90, 45, and 60 degree internal angle. And since the pieces are triangular, there is no issue with all the corners being the same plane. However, even these new sections aren't identical. Using the same nomenclature as before, the Northwest and Southeast sections are mirror images of the Northeast and Southwest ones.

You mean like this: On its way you can note that it passes through this shape: As well as the pentagon-hexagon pattern familiar from a soccer ball.

Here are two side by side: While the original one does have pentagons formed of 5 triangles each, the pentagons don't all fit edge to edge. You can have only two pentagons that do not share any triangles and those pentagons are joined by triangular faces. The one on the right is made up of 12 pentagons that only share edges. if you take it one step further and form these pentagons from triangles like om the left shape, you end up with a solid with 60 faces.

regular tetrahedra won't fit snugly, but irregular ones can. The one on the left still allows you to fit 13 balls in the same sphere as the one on the right, the balls just won't all fit snugly to each other. IF that is a consideration, then the one one on the right is the choice you want. (However, as a D&D die, the shape on the left is better as the one of the right would be "loaded". ) Returning to your original topic, here's another way of dividing up a sphere into 12 identical sections, shown next to the first method.

You shouldn't have to do either. If you load in Pov-ray (I'd try ver. 3.6) and then load Moray, Moray will recognize POV-ray and make the connection for you. Then if you tell Moray to render, it will open POV-ray and start the render. If for some reason, it doesn't, you can do the following: once you've named and saved your Model file, go to the menu bar and hit "render" and then "export". This will create a POV-ray scene file that you can open and render directly in POV-ray. Don't hang your head quite yet. After stepping away for a while and then coming back to it, something started bothering about the whole thing. Then it hit me, neither method could fit into smaller sphere than the other could. Each method involves placing 12 balls surrounding and touching a single middle ball. But that automatically means that the smallest radius sphere that can enclose either group of balls has a radius 1 1/2 that of the balls. So I went back and looked at the my second model and found a small scaling error. It wasn't obvious, but it was enough to throw things off. It looks like its a tie. Here's how the exterior vertices work for both: Your "square and triangle" version and the all triangles version:

Really? Just how old a system do you have? Even if you can't run version 3.7, They still offer 3.6 and that ran well on my last computer, which was 6+ years old when I replaced it a year ago. And if that won't run, they have archived even the older versions. When you say that you can't get the Modeler to run, do you mean that it won't even start-up? Again, it ran fine on my old computer. You need POV-Ray to use it anyway, because Moray uses its rendering engine to actually generate the image. Yes you do have to figure it out for yourself, but it can be simpler when you can see the model visually (top, front, side and camera views) There are also some short cuts you can take by creating groups or unions and duplicating them, etc. Moray is an older modeling program, and since the Pov-ray team took over its licensing hasn't been developed further (its last version was mated to POV-Ray 3.5). There are likely better modelers out there, but I've become comfortable with it and its free. It does have its glitches. Sometimes if you try to copy and paste in a scene that is already complex, it can crash. The nice thing is that since it exports directly as a Pov-ray scene file, you can work around this. I basically can model the different objects in the scene and then assemble the scene through the Pov-ray editor. For example, this is how I assembled this image. Each of the figures were modeled separately in Moray and then assembled in a single scene by editing a Pov-ray scene file. This ability also helps when you want to incorporate some of the features in the newer versions of Pov-ray that Moray doesn't support. (like the new pavement and tile patterns in 3.7) It also had some nice features, like a mesh editor, which allows you to generate a mesh from any object and then manipulate the mesh by points, lines or faces. You can also pan, track, dolly and orbit the camera with the mouse. As far as fitting the spheres more tightly, here is a comparison. On the right is Tar's original method, on the left, a different method both use 13 total spheres. The encompassing boundary spheres around both are of equal size. As can be seen, the one on the left fits entirely inside the boundary, while the one on the right doesn't.

Here's the version you wanted: Upon looking a little further, what this shape represents is a Hexahedron called a triangular dipyramid (basically two tetrahedrons joined at the base) with the "corners" lopped off. Here it is with the corners added back on in a different color to highlight.

I get what you wanted, but both ways are valid solutions, in that the next "layer" fits snug to the layer inside. The alternating square and triangle panels is more of a esthetics issue. I'd post a image of the other solution, as I have it rendered, but my image hosting site is giving me problems right now. POV-ray is the renderand can be found at: http://www.povray.org Moray, which is the modeling program, can be found here: http://www.stmuc.com/moray/

Here's a video of the building of the next "layer":

Rendering software- free Modeling software- free Expertise in usage- priceless! Here's the ping pong figure being built up glass ball by glass ball:

I reset the setting, try it now.

It's a CGI image I created with a 3-D modeling program. Here's an animated version showing a sphere being built piece by piece.

Just for fun, I rendered a version of the sphere divided into 12 identical sections:

at 35 km gravity has only dropped by ~1% ~11 km/sec Even at the "edge of the atmosphere" you are deep in Earth's gravity well. In terms of thrust, anything over 9.8 N would get you there eventually. However, the less the thrust, the more time fuel just spent supporting the weight of the rocket. It is much more efficient to get up to speed quickly, and for that you need more thrust. As far as lift off and getting into orbit goes, solar energy just won't cut it. the solar panels just can't produce the energy to lift even their own weight. Besides, any rocket still needs reaction mass (something to throw to produce thrust). With chemical rockets, the fuel and reaction mass are one in the same. Now there is such a thing as a solar powered ION engine. They produce very low thrusts, so they are only useful once you've gotten into a free fall environment. The advantage of an ION engine is that it has very high exhaust velocity, and the higher the exhaust velocity, the more efficient the rocket (the less reaction mass needed to reach a certain speed). But, as I said, the high efficiency comes with low thrust.

Here's the problem. Launching a rocket to the Moon from a height of 35 km would take 10.73645 kg of fuel per 1 kg of rocket mass, vs 10.8623 kg if launched from the surface (assuming a direct to transfer orbit launch). This is only a savings of ~0.125 kg of fuel per kg of rocket. This also means that for every kg of rocket, your balloon would have to lift 11.73645 kg. The lifting power of Hydrogen when used in a balloon is 1.202 kg/m^3. The density of hydrogen at STP is 0.09 kg/m^3. Thus you would need 0.8788 kg of hydrogen filling your balloon to lift the 1 kg of rocket plus the fuel (this is not even taking into account the weight of the balloon material itself). This is 7 times more than the savings in fuel you would get from launching at 35 km vs. the surface.

The difference is that water is more or less incompressible, while air is compressible. In other words, one cubic ft of water at the bottom of the abyss masses just about the same as 1 cubic ft at the surface. Air is different. Not only does its pressure go down with altitude but so does its density. 1 cubic ft of air weighs less at 100,000 m than it does at sea level. Since the buoyancy of our "balloon" depends on the balloon material weighing less than the volume of air it displaces, as you go to higher and higher altitudes, you have to lower the weight of the balloon to compensate. Thus the thickness( and strength) of the walls of your balloon must also get thinner meaning that they can withstand less and less pressure without collapsing.

Free neutrons decay into a proton, electron and anti-neutrino in a half-life of ~15 min. Besides, a single free neutron actually masses more than a hydrogen atom, so hydrogen gas would be less dense and be more buoyant than a free neutron "gas", not to mention the problems of containing such a "gas".

Moontanman points out the main difficultly in this. When you use a gas, you are equalizing the pressure on the inside and outside, while still having the gas on the inside being of lower density than the outside air. Thus the material from which your container is made does not have to be rigid. If you try to contain a vacuum, you will have a pressure of ~14psi working to crush your container. So let's say that your "balloon" has a 1 foot radius. This gives a surface area of ~1810 in² for a total force of 25334 lbs. This also works out to a volume of 7238 in³. at 20°C, this much dry air weighs about 0.00004 lbs(0.02g). This is the maximum weight that the material of our "vacuum balloon" can have and have even neutral buoyancy. Diamond, for instance, has a density of 3.5g/cm³. A hunk of diamond with a mass of 0.02g would have a volume of 0.006cc. Spread into a shell with a surface area of 1810 in², you get a thickness of 5 nanometers,. This is about 1/40th the thickness of a typical sheet of aluminum foil. Since diamond is 3.6 times harder than aluminum, this hardly seems enough to withstand the 25334 lbs of crushing force exerted by the atmosphere on our balloon.