Tiling a Scaled Pentacube with a Pentacube

Introduction

A pentacube is a solid made of five equal cubes joined face to face. There are 23 such figures, not distinguishing reflections and rotations:

The six blue tiles have distinct mirror images. Kate Jones's names are shown in green. Donald Knuth's names are shown in red.

Here I show which pentacubes can tile a pentacube scaled up by a factor of two or three. If the tile is the same pentacube as that being scaled, the tiling is called a reptile. See Polycube Reptiles.

To see an example of a tiling, click on the # in the appropriate cell of the table.

Disallowing Reflection

These tables show results when the tile may not be reflected.

Scale 2

	SCALE 2
	A	B	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z
A
B
E							#														#
E′							#														#
F
G
G′
H							#														#
H′							#														#
I							#
J
J′
K
L							#												#				#
M
N
P	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#
Q		#	#	#			#			#	#		#			#	#	#			#	#	#
R
R′
S																					#
S′																					#
T
U
V																					#
W
X
Y
Z																					#
	A	B	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z

	SCALE 3
	A	B	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z
A
B
E													#	#
E′													#	#
F
G
G′
H
H′
I							#
J	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#
J′	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#
K
L					#								#	#		#		#					#
M
N
P	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#
Q
R
R′
S
S′
T
U		#	#							#		#	#	#			#	#		#		#	#
V
W
X
Y
Z
	A	B	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z

Allowing Reflection

These tables show results when the tile may be reflected; that is, the tile and its mirror image may both be used in the tiling.

Scale 2

	SCALE 2
	A	B	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z
A
B
E							#														#
F
G
H							#														#
I							#
J											#			#		#		#			#		#
K
L							#												#				#
M
N
P	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#
Q		#	#	#			#			#	#		#			#	#	#			#	#	#
R
S																					#
T
U
V																					#
W
X
Y
Z																					#
	A	B	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z

Scale 3

	SCALE 3
	A	B	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z
A
B
E	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#
F
G
H	#				#	#							#	#
I							#
J	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#
K
L					#								#	#		#		#					#
M
N
P	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#	#
Q
R
S
T
U		#	#							#		#	#	#			#	#		#		#	#
V
W
X
Y
Z
	A	B	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z

Last revised 2020-04-23.

Back to Polyform Tiling < Polyform Curiosities

Col. George Sicherman [ HOME | MAIL ]