According to traditional Japanese beliefs, the tortoise is a haven for immortals and the world mountain, and symbolizes longevity, good luck, and support. The tortoise is a favored motif by netsuke-carvers and other artisans, and is featured in traditional Japanese wedding ceremonies.
In Japan, the minogame is so old it has a train of seaweed growing on its back, is a symbol of longevity and felicity. A minogame has an important role in the well-known legend of Urashima Tarō.
There is also a well-known artistic pattern based on the nearly hexagonal shape of a tortoise's shell. These patterns are usually composed of symmetrical hexagons, sometimes with smaller hexagons within them.
Photo of minogame: http://art.thewalters.org/images/art/large/l_pl9_541355_gp_bw-2.jpg
Photo of netsuke: http://collecting.wdfiles.com/local--files/image:turtle-hatching-netsuke/netsuke.jpg