polycubes

(a "flat" polycube derives a solution from the corresponding polyomino)

tetracubes

2
+--+--+
|     |
+     +
|     |
+--+--+
2 levels deep


2
+--+--+     +--+--+
|u    |     | d|  |
+  +--+     +--+  +
|  |u |     |    d|
+--+--+     +--+--+
level 1     level 2


2
+--+--+     +--+--+
|   u |     |  | d|
+  +--+     +  +--+
|  |u |     |    d|
+--+--+     +--+--+
level 1     level 2


pentacubes

2  (F pentacube)  without hole
+--+--+  +     +  +--+--+     +--+--+  +
|   u |           |ud   |     |  | d|
+--+--+  +     +--+--+--+     +  +--+  +
|u |           |ud|           | d|
+--+  +  +     +  +  +  +     +--+  +  +
               |  |
+  +  +  +     +--+  +  +     +  +  +  +
level 1        level 2        level 3


2  (U pentacube)
   +--+        +--+
   |u |        | d|
+--+--+--+  +--+  +--+
|u     u |  | d|  | d|
+--+--+--+  +--+  +--+
   |u |        | d|
   +--+        +--+
level 1     level 2


3  (V pentacube)  without hole
+  +--+  +     +  +--+  +     +  +--+  +
   |u |           |ud|           | d|
+  +--+  +     +--+--+--+     +  +  +  +
   |  |        |        |        |  |
+  +  +  +     +--+--+  +     +  +  +  +
   |  |              |  |        |  |
+  +  +  +     +  +--+  +     +  +--+  +
   |u |           |ud|  |        | d|
+  +--+  +     +  +--+--+     +  +--+  +
level 1        level 2        level 3


3  (W  pentacube)
+  +  +--+--+     +  +--+--+  +     +--+--+  +  +     +--+  +  +  +
      |u    |        |u   d|        |u | d|           | d|
+  +--+--+--+     +--+--+--+  +     +  +--+  +  +     +--+  +  +  +
   |     |        |u |              | d|
+--+  +--+  +     +  +  +  +  +     +--+  +  +  +     +  +  +  +  +
|     |           |  |
+  +--+  +  +     +--+  +  +  +     +  +  +  +  +     +  +  +  +  +
|  |
+--+  +  +  +     +  +  +  +  +     +  +  +  +  +     +  +  +  +  +
level 1           level 2           level 3           level 4


4  (T pentacube)
+--+  +  +  +     +--+--+--+--+
|x |              |x |        |
+--+  +  +  +     +  +--+  +--+
                  |  |  |  |  |
+  +  +  +  +     +  +  +  +  +
                  |  |  |  |  |
+  +  +  +--+     +--+  +--+  +
         |x |     |        |x |
+  +  +  +--+     +--+--+--+--+
levels 1 and 3    level 2


2
   +--+        +--+
   |u |        | d|
+--+--+--+  +--+  +--+
|        |  |        |
+--+  +--+  +--+--+--+
   |u |        | d|
   +--+        +--+
level 1     level 2


2
   +--+        +--+
   |u |        | d|
+--+--+--+  +--+  +--+
|u       |  | d|     |
+--+  +--+  +--+  +--+
   |  |        |  |
   +--+        +--+
level 1     level 2


2
   +--+        +--+
   |u |        | d|
+--+  +--+  +--+--+--+
|u    |u |  | d|    d|
+--+--+--+  +--+  +--+
   |u |        | d|
   +--+        +--+
level 1     level 2


2
+  +  +--+  +     +  +  +  +  +
      |  |
+  +--+  +--+     +  +  +--+  +
   |   u    |           | d|
+--+--+--+--+     +  +--+--+  +
|   u    |           | d|
+--+  +--+  +     +  +--+  +  +
   |  |
+  +--+  +  +     +  +  +  +  +
level 1           level 2


2
+--+  +--+     +--+--+--+
|  |  |u |     |       d|
+  +--+--+     +--+--+  +
|u       |     | d|  |  |
+--+--+--+     +--+  +--+
level 1        level 2


2
+--+  +--+     +--+--+--+
|u |  |u |     | d|    d|
+  +--+  +     +  +--+  +
|u    |u |     | d|  | d|
+--+--+--+     +--+  +--+
level 1        level 2


2 not allowing reflections
+  +--+--+     +  +--+  +     +--+--+  +
   |u    |        | d|        |     |
+--+  +--+     +  +--+  +     +--+  +--+
|     |           |u |           | d   |
+--+--+  +     +  +--+  +     +  +--+--+
level 1        level 2        level 3


2
+  +--+  +  +     +--+--+  +  +     +  +  +  +  +
   |u |           |  | d|
+  +--+  +  +     +  +  +--+--+     +  +  +  +  +
                  |  |        |
+  +  +  +  +     +  +--+--+--+     +  +--+  +  +
                  |   u |              | d|
+  +  +  +  +     +--+--+  +  +     +  +--+  +  +
level 1           level 2           level 3


2 allowing reflections
+--+--+--+     +--+--+
|u |     |     | d   |
+--+  +--+     +--+  +--+
|u    |        | d|     |
+--+--+        +--+--+--+
level 1        level 2

4 not allowing reflections
+--+--+--+--+--+     +--+--+--+--+--+
|u |     |   u |     | d   |     | d|
+--+  +--+  +--+     +--+  +--+  +--+
|u    |     |u |     | d|     |    d|
+--+--+--+--+--+     +--+--+--+--+--+
level 1              level 2


2 allowing reflections
+--+           +--+
|u |           | d|
+  +           +--+
|  |           |  |
+  +--+--+     +  +--+--+
|     |u |     |       d|
+--+--+--+     +--+--+--+

4 not allowing reflections
+--+--+  +     +--+--+  +     +--+--+  +
|u    |        |ud|u |        | d| d|
+--+--+--+     +--+--+--+     +  +  +--+
|      u |           | d|     |  |  |  |
+  +--+--+     +  +  +--+     +--+  +  +
|  |   u |           |ud|     |     | d|
+--+--+--+     +  +  +--+     +--+--+--+
level 1        level 2        level 3

(second tiling)
+--+--+  +     +--+--+  +     +--+--+  +
|u |u |        | d|ud|        |    d|
+  +  +--+     +--+--+--+     +--+--+--+
|  |  |u |           |ud|     |  |    d|
+  +--+  +     +  +  +--+     +  +--+--+
|     |  |           |u |     |       d|
+--+--+--+     +  +  +--+     +--+--+--+
level 1        level 2        level 3



2 allowing reflections
+  +--+--+  +     +  +  +  +  +
   |     |
+  +  +--+  +     +--+--+--+--+
   |u |u |        |    d| d   |
+  +--+  +  +     +--+--+--+--+
   |     |
+  +--+--+  +     +  +  +  +  +
level 1           level 2

4 not allowing reflections (has hole)
+  +--+--+  +  +     +  +  +  +  +  +
   |     |
+  +  +--+--+  +     +--+--+  +--+  +
   |u |   u |        |    d|  | d|
+  +--+--+--+  +     +--+--+  +  +--+
                     |     |  |     |
+  +--+--+--+  +     +--+  +  +--+--+
   |u    |u |           | d|  | d   |
+  +--+--+  +  +     +  +--+  +--+--+
      |     |
+  +  +--+--+  +     +  +  +  +  +  +
level 1              level 2

5 not allowing reflections (no hole)
+  +--+--+  +  +     +  +  +  +  +  +     +  +  +  +  +  +
   |     |
+  +  +--+--+  +     +--+--+  +--+  +     +  +  +  +--+  +
   |u |   u |        |    d|  | d|                 |  |
+  +--+--+--+  +     +--+--+--+  +--+     +  +  +--+  +  +
                     |     |u |     |           | d   |
+  +--+--+--+  +     +--+  +  +--+--+     +  +  +--+--+  +
   |u    |u |           | d|  | d   |
+  +--+--+  +  +     +  +--+--+--+--+     +  +  +  +  +  +
      |     |
+  +  +--+--+  +     +  +  +  +  +  +     +  +  +  +  +  +
level 1              level 2              level 3