truthseeker

The answer to your question is contained in your post. With 2D polygons (all the same size and shape, if you want) you can construct 3D polyhedra. Then from those 3D polyhedra you can construct 4D polytopes, and from there 5D, etc. One of the more interesting features of higher dimensions is the fact that surface areas and corners become much larger than bulk volumes. Of course you may say that's nonsense, since areas and volumes have different units, so how can I compare them. But it is true nevertheless in many different senses. For instance, the ratio of the surface area of a unit sphere to its volume starts to decrease above 7 (if I remember correctly) dimensions. Nearly all the volume of a higher dimensional cube is in its corners. The number of corners grows much faster than the number of faces, so most vectors point towards corners rather than towards faces, etc. This is particularly apparent when you study polytopes in high numbers of dimensions.

Not sure I really want to get involved in this discussion, but I think there are two questions that have to be separated: (1) Is evolution by natural selection the origin of the species we see around us now, and in the fossil record? That was Darwin's idea and it is the unifying principle of modern biology (to the extent there is one) (2) Does evolution happen sometimes? Here "evolution" can refer to everything from evolution of populations of organisms to computer algorithms that employ a form of selection to optimize something. The answer to (2) is indisputably "yes". The answer to (1) is "yes" as well but there's much more room for argument because it's a statement about the past. As for entropy and the second law, the fact that (2) happens immediately tells you that evolution is consistent with the laws of physics, so you cannot object to (1) on those grounds.

OK. If the air was blowing from the right end of the tunnel towards the left, you are predicting the cart will move to the left slower than the air, so slower than 10mph (steady-state). Now suppose we turn off the fans and turn on the treadmill belt with the belt moving at 10mph. The belt is moving from the left towards the right. Instead of running the cart up to 10mph and then releasing it, the arm holds the cart in place (at rest with respect to the air) and then lets it go. This is the scenario in the youtube video I linked to above. Do you agree that the cart will accelerate and advance up the belt - to the left - at some non-zero steady-state speed? (This is just as shown in the video I linked to above.) If you agree with that, note that after a trivial Galilean transform (to a frame moving at 10mph with respect to the original frame) the air and belt velocities are identical in the two scenarios, as is the initial condition (cart at rest with respect to the air) - but you are predicting a different outcome for the forces and motion of the cart from the same initial condition. If I take your prediction for the fan-driven case and transform so the air is at rest and the belt moves at 10mph from left to right, your prediction is a cart that moves to the right, losing ground against the belt. On the other hand the video shows a cart moving to the left, up the belt. How then is your prediction consistent with Galilean invariance? The only real difference between these two experiments is what is ultimately creating the relative velocity between the air and belt (it's a fan in the first case and a treadmill belt motor in the second). How is the cart supposed to know about that, when the air and belt (the two things it interacts with) behave identically after the Galilean transform?

Let's make this very simple. Suppose I take a treadmill cart and put it in a very long wind tunnel. The floor of the tunnel is a treadmill belt, but the treadmill's motor is off. The wind tunnel fans are on and create a steady wind at 10 mph. I attach the cart to a moving arm that holds it and runs it down the tunnel at 10 mph (at rest with respect to the air) for a few seconds and then releases it. Question - will that cart accelerate to a steady-state speed faster than 10mph?

Whether or not the cart can self-start on a (sufficiently long) treadmill is irrelevant to the claim of whether it can travel ddw faster than the wind steady-state. Self-starting (i.e. starting at rest with respect to the ground/treadmill belt) is a different and transient regime compared to ddw and how the cart behaves under those different conditions depends on its design. Starting a treadmill cart at rest with respect to the air is Galilean equivalent to towing an outdoor cart up to windspeed and then releasing it. If the cart accelerates downwind from that initial condition and maintains faster than wind speed (as both the treadmill carts and the Blackbird can do) that proves the claim. So for the main point at issue there's no reason to consider self-starting. With that said, here are two videos. One shows a cart self-starting on a circular treadmill. At around 2:15 the other video shows a cart on a regular treadmill where the cart is pushed back and then accelerates up the belt. While it is moving back it is experiencing a tailwind (falsifying your assertion that this never happens). This shows that you really do not understand GTs, or maybe wind, or maybe both. There is no relevant difference between the tailwind on a treadmill compared one outside or over an ocean or lake. The air is moving relative to the treadmill surface - that is wind, full stop.

What part of "wind is the relative motion of air over a surface" are you having trouble grasping? When you ride a long moving walkway you feel a wind in your face (which for the cart would be a tailwind, since it faces up the belt). If you imagine that walkway is really wide as well as really long, you'd have no way of knowing if it's the belt that's moving and the air that's at rest, or the air that's moving and the belt that's at rest. The two are equivalent by a Galilean transform. You really can't get your head around Galilean transforms, can you? You're completely stuck in the frame where the air is at rest. Again, that's simply not true. If you like you can start with the cart moving with the belt and watch it self-start until it's going up-belt. Again - if the belt is really big, you cannot distinguish between a "real wind" - air moving and belt at rest, where for instance the air motion is produced by a fan and the belt is off - and "belt wind" - where the air is not moving but the belt is. You cannot distinguish by any experiment because they are Galilean equivalent. Since we know the cart advances up the belt in the latter scenario, we therefore know it will advance downwind faster than the wind in the former.

Do you really not see the problem there? Wind is the RELATIVE motion of air with respect to a surface (the belt in this case). "Powered by the wind" means powered by that relative motion, which is exactly what happens for the cart (which would obviously not work if air and belt were at rest with respect to each other, for instance if the treadmill is off). Imagine a huge treadmill inside a giant wind tunnel, so wide and long you can't see the edges or ends of it. By Galilean invariance there is no experiment that can distinguish between that treadmill belt moving and the air at rest, or the belt at rest and the air moving. Therefore if the cart can advance up the treadmill belt, it can also advance ddw faster than the wind. If not, that's an experiment that can distinguish these two scenarios, and that is impossible by Galilean invariance. (The cart was tested on a small treadmill, but I assume you're not going to claim the results would be different on a large treadmill?)

I completely agree with this. No, I'm not. I said explicitly that the treadmill cart is not outside. Wind, by definition, is relative motion of air or gas with respect to a surface. That surface doesn't have to be "the ground". It can be the ocean's surface, or a layer or Jupiter's atmosphere, or the belt of a treadmill. The fact that you don't understand this shows that you have not understood Galilean relativity. Precisely. That's why there is in fact wind with respect to the treadmill belt, just as there is wind outside when the air moves with respect to the ground or ocean. That depends on what the cart's state of motion is. In its steady-state it is moving directly downwind faster than the wind and so it feels an apparent headwind. To establish that the cart can move ddw faster than the wind (steady state) simply requires the observation that it advances up the belt (steady-state). That is not just EVIDENCE for ddw faster than the wind, it IS ddw faster than the wind. This does not prove that the Blackbird can go ddw, or that the treadmill cart would necessarily work under any conditions other than the ones actually tested, but it does prove that ddw is possible and therefore cannot violate conservation of energy or any other law of physics.

The cart advancing up the treadmill is Galilean equivalent to (so exactly the same as) the same cart going directly downwind faster than the wind. Its not doing it outside and it's very small and light and the wind is perfectly steady, but it's doing it.

Discussing a physics topic that's been around for decades, is the subject of two recent Veritasium videos with over ten million views, reveals interesting features of reference frames and energy, and has been used as a physics olympiad test question in multiple countries, is off-limits? That's bizarre.

From personal experience, running on a treadmill set to some speed is definitely easier than running outside and averaging that same speed. My guess is it mostly has to do with the "tailwind" and how elastic the treadmill surface is. Your feet spring off it in a way that's pretty different from pavement, and that might save significant energy. Maybe the fact that it's exactly the same all the time and you don't have to watch where you're going (I typically run outside on busy streets or tracks with lots of other people) helps too.

Yes it means that, there's no such thing as absolute rest. A turbine in a wind tunnel vs flying through the air? Yes, it would generate the same power if it sees the same apparent wind (speed of the air relative to the turbine).

It's called Galilean relativity because it's been understood since the time of Galileo 400 years ago that there is no such thing as absolute motion. The only thing that can have any physical effect is relative motion. Wind tunnel data is just fine.