The 10000 sided die is about the size of a tennis ball and looks round. It is round. Numbers are very small and read by a laser device which is pushed into position over the dice after it stops rolling. The (self-levelling) laser emitting device shines straight down, and detects only reflection directly
back upwards - numbers not really "on top" reflect slightly off to one side.

How about rolling the hyperdie (two
dice, one die, by the way. Although
oddly the same rule does not apply to
mice) on a painstakingly-levelled glass
flat-bottomed bowl filled with an
optically-dense black liquid? Then, if
the liquid is opaque enough, you'll be
able to look up from under the bowl
and see only the single micro-face
which is resting on the glass; the others
will be obscured by the intervening
liquid.

This would work on
the same principle as those "prediction
balls" where an icosohedral (?) die floats
up to the top of a glass jar full of black
liquid.

Also, is it possible to
make a convex object with exactly
10,000 identical faces? The rules of
geometry might restrict you to, say,
12,436 or 9,084.

What if it didn't have any numbers at all,
but transmitted the bottom-facing
result though bluetooth to another
device. The problem of course would
then be in assuring total equality of
balance, with a powered bluetooth
transponder in there.

Unless you have a supercomputer
nearby to work out a random number
and transmit it to the bluetooth
transponder during the time the dice is
rolling. This number can then be
transmitted as the result of the dice
roll.

Make a hollow sphere out of crystal glass and engrave numbers on the inside. The sphere contains water and a floating, but snugly fitting, second sphere. The second sphere is free to move but contains a laser pointer. The weight distribution in the second sphere causes the laser always to point upwards.

Roll the die and wait. The number will be projected onto the ceiling.

I should point out that each number is cunningly engraved with a suitable lens to cause a magnified image to be formed.

From Mathworld ("Polyhedron" (linked), and others from that):

There are a total of nine regular polyhedra: five Platonic solids, with a maximum of 20 faces for the icosahedron and four Kepler-Poinsot solids, with a maximum - sort of - of 53 faces for the great icosahedron. You might find it hard to roll that one, though.

Getting into the semiregular polyhedra - or Archimedean solids - of which there are 13, the snub dodecahedron has the most faces with 92.

Johnson solids, which have regular faces but are not regular polyhedra, max out at 62 faces (J71 to J75, my favourite of which is J73, the parabigyrate rhombicosidodecahedron).

Of course, you could also consider that the numbers could be placed at the vertices or the midpoints of the edges.

A D100, is two tensided dice with the one going from 1 to 00 and the other has 1 to 0. Roll 00/1=one. Roll 00/0=one hundred. A similar trick could be used with the five dice.

And I know I just couldn't resist counting how many times I have to roll this super die before I get 10,000. And then I would have to break that record, and then, etc.+

I like the "magic 8 ball" adaptations by [DenholmRicshaw] and [Basepair]. [Hippo], there will be no more of your futuristic supercomputer random number generating flimflam. These are dice we are talking about.

[Basepair] I too suspected that there might only be certain numbers which lend themselves to even distribution over the surface of a sphere. This is the sort of thing those math people figure out for a living.

Surely to all intents and purposes this would be round. meaning that it would take a long amount of time to stop if you were to give it anything more than a gentle tap.

I wondered how long it would take to align the reading device so that it got a read.
[hippo] Did you mention a supercomputer because cheap calculators' random number generators are not truely random? No wonder I haven't yet won the lotto. I'm still looking for someone to invest around $8 million USD on a 50% chance of winning scheme. I think I have charts & graphs on my old computer.

I think aligning the device would be a problem. Also chasing the thing all over the table top. Perhaps it should have a special saucer with a tiny level sweet spot in the very center. Eventually it would settle there. The reader would be ready.

A few modifications would make the die more interesting AND more readable. 100-sided die aren't difficult to read and they are maybe 1.5 inches diameter. I believe a 6 inch diameter die would provide the right balance between readability and size.

Some can see the number with the naked eye, some need a magnifying glass.

But, to reduce the amount of numbers you have to put on that tiny little face, a color code for the final digit can be used. I suggest a modified ROY G. BIV type thing.

Dark, blood red = 0
Bright red = 1
Reddish Orange = 2
Yellowish Orange = 3
Yellow = 4
Yellowish Green = 5
Green = 6
Bluish Green = 7
Blue = 8
Violet = 9

the side would be numbered 0-9999, but each red,blue, etc pane would only be numbered 0-999. Thus, you would only need to fit 3 digits into that tiny space, and the color would be an indicator of your general luck.

A hollow crystal sphere (but with thick walls to avoid breakage), with one side flattened, mathematically calculated to stop the sphere from rolling after so many revolutions (on the average) after a toss, so that the thing tends to stay on the table. When the electronic innards detect that the thing has come to a stop, it generates the number pseudorandomly between 1 and 10,000 and displays it with one of those cool wand display animators they use now on some alarm-clocks/message centers, so that the number animates in blue digits, rotating around so that everyone at the table can see it. Make it as small as the electronic innards can cheaply fabricate (and so that the number generated is legible without squinting). Power to run the thing is generated by motion--when you shake it up in your hand, you run a little counterweighted device of the sort they used to use in watches to charge up the microcell a little.