A 3D-continuous cube-spiral

UPDATE, May 3rd, 2021

The hereunder technique works... for a half-space only! It is worth reading, though, as the basic principles remain the same for the 5x5x5 solution-cube.

To "spiral" continuously the full 3-space you cannot use a "basic" 2x2x2 cube, or a 3x3x3 cube as both will cover only a half-space: it is impossible, for instance, to start in the middle of a 3x3x3 cube and visit its 27 unit-cells continuously (and Ulam-like style). But it is possible starting from the very center of a 5x5x5 cube as discovered last night: see the picture at the end of this post. (A 4x4x4 was not -yet- fully searched by me).

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Start your journey somewhere in an elementary cell of the 3-D space, facing South [which is one of the 6 possible directions we will use here: N, E, S, W, U(p) and D(own)].
You will now enter a “Blue dice” (which is cut in 3 slices – bottom, middle and upper) through a “red door” located exactly in the middle of one of the 6 blue faces.
Start your walk by following the blue arrows and change slice when requested. You will exit the blue dice through a “blue door” (in the middle slice) that is 90° to the red door:

(Note the two nice clockwise “spirals” that occur in the bottom and upper slices.) Your 27 successive “blue movements” are listed hereunder: the 1^st letter is the direction you are coming from and the 2^nd letter is the direction you are going:
1. NE (“coming from the North, going East”),
2. WD (“coming from the West, going Down”),
3. US (“coming from Up, going South”),
4.NS / 5.NW / 6.EW / 7.EN / 8.SN / 9.SE / 10.WS / 11.NU / 12.DU / 13.DN / 14.SE / 15.WS / 16.NS / 17.NW / 18.EW / 19.EN / 20.SN / 21.SD / 22.US / 23.NS / 24.NE / 25.WE / 26.WN / 27.SE

What comes next?
Next comes... recursion!
Consider indeed the 27 cells arranged in our 3x3x3 (blue) cube as a greater, unique cell belonging to a greater unique cube. We will pace this greater cube using a master-plan we already have: the blue path inside the blue dice!
This means that we must leave the 1^st Blue dice and enter a 2^nd one – the one that corresponds to the number 2 here:

After visiting the 27 squares of the Blue dice #2, we will leave this dice and enter a 3^rd one: the one that belongs to the “slice” below. From there we will visit the 9 dices of the “bottom slice”:

Is it easy to use the master-plan we have? Yes – but there is a trick: the “blue master-plan” does not explain how to walk through some dice! On the figure above we see that the dice #4 will be crossed between the dices #3 and #5. It will be the same later with the dices #6 or #8.
The “blue master-plan” was ideal to make a step forward, then a 90° turn, then another step. But we need another masterplan to go straight (in other words: is it possible to enter a dice using the same red door as before, but exiting the said dice using the cell exactly opposite? The answer is yes: the “Green dice”.

We see that the visit starting at the red door ends behind the green door, exactly opposite. (Note again the two nice clockwise “spirals” that occur in the bottom and upper slices.)

The list of the successive 27 “green movements” is now:

1.NE / 2.WS / 3.NS / 4.ND / 5.UW / 6.EW / 7.EN / 8.SN / 9.SE / 10.WE / 11.WS / 12.NW / 13.EU / 14.DU / 15.DN / 16.SE /17.WS / 18.NS / 19.NW / 20.WE / 21.EN / 22.SN / 23.SD / 24.US / 25.NS / 26.NE / 27.WS

Ok, we are now fully equipped with our two tools, the turn dice, and the pierce dice.

Time to “spiral” the 3D space now. In the left hand we have the “blue dice-masterplan”, in the right one we have the “green dice-masterplan”. We will use those when we must make a turn or go straight.
At the "smallest" level, the level #1, we have paced the 27 cells of the blue dice.
We will now attack the level # 2 (and its remaining 26 dices for a total of 27 x 9 = 243 cells).
We will exit the dice #1 and enter the dice #2, then go down one “slice” to explore the 9 dices of the “bottom slice”:

Entering the blue dice #2 is easy, as there is only one “red door” per dice – but what about the orientation of the #2 dice? We must exit the dice in direction Down. The little arrow between the dices #1 and #2 (above) indicates that the #2 dice will currently direct us towards North (after crossing its 27 cells), instead of Down. We must thus turn 90°counterclockwise the dice #2 (from the point of view of the little arrow we have mentioned). The big hand drawn digit “2” (above) will thus no longer be visible, but we will at least exit in the Down direction we wanted!
The dice #3 is a blue one too – as it must turn us from Up to South. Its “red door” is on the face with the big hand drawn “3”, thus its exit door is in the wrong direction, pointing towards “10” instead of “4”. We must then turn also the “3”-dice counterclockwise (from the point of view of the “2”-dice on top of it). The hand drawn “3” will thus look like a “m” on the above photography, instead of a “3”, but we will at least exit its 27^th cell in the South direction we wanted!
The dice #4 is a green one (as we do not want to turn, just go straight): we simply place its red door pointing towards North and that is it.
The dice “5” is a blue one (we must turn 90° to cross the dice #6); again, we place its red door facing South and turn the dice 180°: its exit door is now oriented toward the dice #6.
The dice #7 is a green one. We just must put its red door facing East.
Etc.
(to be continued)

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The 5x5x5 plans: starting cube and "tool" cubes

UPDATE of the UPDATE (May 7th, 2021)

Half-space is not enough, of course — see our last attempt in detail here for the full lattice.