orthogon wrote:I left it going overnight in the end. No idea how long it took:
I got the range a bit wrong so it made about 25 frames in which there was no intersection at all! I might do a smoother one now I know it works. Note that this one is at a more general angle than the case Pfhorrest was talking about. I guess that case was at 45 degrees in the w-x, w-y and w-z directions? I might have a go at that case.
So I worked out why I'd overcooked the range so much. The hypercube has sides of 1 unit, so the squaw on the hippopotamus is 4. This means the diagonal is 2 (that's a neat thing about 4D!), and since it's centred on the origin it can only extend +/- 1 unit in any direction, so I only needed to explore the range -1<=w<=+1. I think this is Pfhorrest's case where the hypercube's edges are 45 degrees from all three axes of the slice (x,y and z):
For some reason Mathematica decided to get really really slow, so I didn't go to a step size of 0.05 as I'd intended. I'm still not sure I got the angle right, since it doesn't seem to be quite as Pfhorrest described it, and I was expecting it to be symmetrical about the origin. What I did was: rotate by 45 degrees between the w and x axes, then by 45 degrees between w and y, then between w and z. I really can't work out whether this is the right thing to do (I'm not even sure in 3D) and if anyone can advise I'll happily set it off overnight doing whatever it should have been!
I have to admit to a certain amount of visceral fear and trepidation while I wait for these things to appear on my screen. My hairs stand on end and I feel unusual. It seems that I find the whole idea of the 4D hypercube really unnerving. It's like a kind of Eldritch Abomination, a horror beyond my comprehension of which my feeble brain can only comprehend an infinitesimal part at any one time. Is it just me?
xtifr wrote:... and orthogon merely sounds undecided.
I have to admit to a certain amount of visceral fear and trepidation while I wait for these things to appear on my screen. My hairs stand on end and I feel unusual. It seems that I find the whole idea of the 4D hypercube really unnerving. It's like a kind of Eldritch Abomination, a horror beyond my comprehension of which my feeble brain can only comprehend an infinitesimal part at any one time. Is it just me? orthogon
Posts: 743 Joined: Wed May 16, 2012 11:52 pm UTC
umm. Apparently, it is Not, just, you.
Are you serious? Or; Is what you wrote, you being silly?
two little tinny thoughts about one Huge idea. 1. Why would you be afraid to know The Truth?
A monster waits? Through The Membrane, a monster awaits, for you? eeewww.
2. Do you Remember The Mirror discussions? Some silly thing about the world, Through The Looking Glass.
I saw a reflection. It Stopped me. I thought of this Thread.
I am not good at math. I can not describe it with math.
A person does not need to be good at math to know fire is hot. A persons does not need to be good at math to know a lot about fire.
The kind of knowledge I have is like a person that builds a fire, pretty much, every day, But, never had a chemistry class.
Mirrors do some weird stuff to light, sometimes. Well….? You are frightened of the Fifth Dimension? What did you put in there?
2. a) Have you ever noticed the reflection of a tree, in water. Sometimes; Often, I find the reflection to be a more engaging vision than the tree, its self. You?
What is different about a building, a tree, a statue when reflected in water? The building and the statue look more, alive. The tree Is alive. right?
How can a living thing, look more alive?
Wake it up?
It is darned hard to prove. I don't like it when some of that stuff, proves its self.
Brother Void... That is a funny man. I don't understand it all. Some of it 'Speaks' to me.
The Kafka of Today. (tisk, tisk what are we reading, now?)
It takes faith to believe, and it takes courage not to, and who is to say which is the deeper and more truthful. --Herbert Weisinger
You often hear about believers who have a crisis of faith, but what of the skeptics among us who have a crisis of doubt? For years we skeptics have decisively refuted the metaphysical claims of the great religions and scoffed at the pretensions of newfangled spiritual fashions. But then our doubt is suddenly shaken by an unbidden mystical experience. The power of this direct cognition of ultimate reality, beyond word or image, is undeniable. But does it prove the existence of God? If you remain skeptical you find yourself in a difficult state. You now seriously doubt your doubt and yet have no abiding faith to replace it. How do you proceed? You can no longer be atheistic because you've communed with the divine. You can't be religious because the existence of God is still in question; what's more, religious representations of God now get in the way of your direct mystical experience. Nor can you be agnostic because you're far from neutral on the subject. You must become a skeptical mystic. As you cut your own singular path to the great whatever, you must now treat your own experiences with the relentless skepticism you once reserved for the claims of others.
I am One with a God I do not believe in.
I think it is funny. You?
To doubt your doubt. In Science doubting our doubt was often understated.
Who doubts their doubt? What if it is more complicated?
I do not do complex. It's too hard. Simple is difficult, too.
To Live a Simple Life is Man's Most Complicated Task.
Life is, just, an exchange of electrons; It is up to us to give it meaning.
We are all in The Gutter. Some of us see The Gutter. Some of us see The Stars. by mr. Oscar Wilde.
Those that want to Know; Know. Those that do not Know; Don't tell them. They do terrible things to people that Tell Them.
So it turns out that three successive rotations by 45 degrees each does not give the result I wanted, so my previous plot was indeed not Pfhorrest's case either. However I realised that I could get there in one step by using RotationMatrix to find the transformation that rotates a diagonal of the hypercube (1 1 1 1)T so that it lies in the direction (1,0,0,0)T, i.e. parallel to the w axis (the dimension that we're representing using time). I also added a bit of camera movement while I was at it, and here is the result:
Again, I must congratulate Pfhorrest on an eloquent description of something that ought to have defied description!
@addams: yes, I was being serious about feeling freaked out by it, and yes, I think I felt the same in the mirror discussion. @Klear: Five dimensions? You're just making it worse. At least the hypercube doesn't have any tentacles...
xtifr wrote:... and orthogon merely sounds undecided.
Oh awesome, that is exactly what I was visualizing in my mind! Thank you orthogon for actually doing the hard math and rendering to confirm my intuitions.
I also really like seeing the other cases you rendered, which I hadn't visualized so thoroughly in my mind. It's cool how they really give more of an impression that something like a cube is in some way passing through something, than my corner-on case does.