45 Degrees of NonsenseI think we can all agree that imparting the *least* amount of energy to the dice such that chaotic motions are minimized is ideal for the DI.
But does this mean that they need to land with a "45 degree angle"? Not necessarily.
Landing at 45 degrees simply means the forward and downward force vectors are equal.
Now it is true that as the angle deviates from 45, that the sum of the forward and downward forces increases to get the dice to land the same distance away (sqrt(down^2 + forward^2).)
But in addition to the total force, we need also be concerned with how the downward force and the forward force is dissipated. Or more to the point, how the table will interact with the downward and forward forces.
Try this as an experiment: take one die and hold it 1" above and parallel the table. And drop it. At 1" the top face should most likely remain the top face.
Now try 6". At 6" the dice is probably much more likely to rotate by 90 degrees (or more) such that a side face becomes the top face.
Now try a foot above the table. Bounces alot, huh? Remember that falling object accelerate at the rate of 32'/sec^2. Thus at 1", the die is traveling at 28 inches per second at impact, dropped from 6" it travels at 68 inches per second, and from 12 inches it travels at 96 inches per second. (velocity = sqrt(2gh).
The kinetic energy is 1/2 the square of speed * mass, so for simplicity the mass of the die = 1 and the energy at 1" drop is 392, at 6" it's 2312, and at 12" it's 4608. The kinetic energy increases rapidly even for a few inches off the table.
For the dice, the downward force vector is always a result of the distance between the table and the PEAK of the toss (the zenith of the toss arc). Assuming the dice go up after leaving the shooter's hand, then the downward force is purely a result of gravity. This also means that the downward velocity is constantly accelerating.
Now let's look at the forward force vector.
if you lower the peak of your toss, you have to increase the forward velocity to get the dice to the same location.
For the sake of simplicity, let's assume that the zenith of your toss happens 4' from the target landing area.
If the peak of your toss is 1" off the table, the dice need to be traveling at 55' per second or 38 miles per hour to touch first at roughly 8 feet away from your throwing position. That's about 24 times faster than the downward velocity.
On the other hand if your zenith is 24" off the table, then your forward and downward velocities at target impact are about equal - 11 feet per second or a bit less than 8 mph.
If your zenith is just 12", then the forward velocity would need to be about twice the downward velocity - 16' per second forward vs the 8' per second downward velocity at impact.
Now that we understand the relationship, how do we use this information?
In the experiment at the beginning of this article, we saw that table bounce characteristics can greatly randomize the die even with zero forward velocity.
Table bounce is directly related to the downward velocity.
Sliding (forward velocity with no downward velocity) is considered cheating because it results in such a huge advantage to the shooter.
However, extremely high forward velocities leave a huge amount of kinetic energy to be dissipated, and we really want the dice to rotate and undulate as little as possible as they come to rest.
It is ultimately a question of balance - and one that is different depending on table conditions. The balance is one of reducing bounce due to downward force, vs. reducing chaotic roll outs due to excessive forward velocity.
Enter backspin and dice phase
Let's first review gyroscopic precession. As we may recall, a force applied perpendicular to the rotational plane of a gyroscope (a spinning object such as a die with backspin), will have the resultant force 90 degrees ahead in the direction of rotation. Or put another way, precession states that when a force from the outside tries to tilt a spinning gyro, that force is actually felt by the gyro as if it had been applied at a point 90 degrees away, in the direction of the rotation. In other words, “the cause precedes the effect by 90 degrees.”
This explains some of the kooky gyrations you may see in slow motion photography of dice landing. If a die with a great deal of spin touches at a corner (which is outside the center of inertia), the die will yaw off in a new direction.
As long as the die is spinning on a specific axis though it should continue to spin on that axis - but that axis may no longer be parallel to the table surface, so when the spin dies down, the die can easily roll out "off axis".
This makes it difficult to determine if your "primaries" are on the X (pitch) or Y/Z (roll/yaw) axes.
Keeping in mind that 4 out of 6 primary yaw outcomes are exactly the same as the 4 primary pitch outcomes, how can you tell? It's difficult to affirmatively determine without slow motion photography. What you *can* tell are the dice "relative phase" results - that is, are the dice more or less likely to land in phase, or 90 or 180 degrees out?
If you are more successful at keeping the dice together, and working in relative phase, you can choose a dice set that works well relative to your phase response.
Scatter reduces control and increases randomization of dice. Scatter is a big problem with table bounce so minimizing table bounce minimized bounce related scatter. And minimizing bounce means reducing the downward velocity.
Scatter is also a problem when hitting the back wall too hard, so ideally you want the forward velocity to dissipate by then.
What I've been experimenting with is a toss with a low zenith, landing farther away from the wall, so that the added forward velocity is dissipated in a long roll out.
This *seems* to be giving me good relative dice phase results. I'm curious if anyone else throws this way, and what your results are?
I would say that it appears that the concept that the dice *must* land at 45 degrees is nonsense....
Dances With Dice