Professor Kenji Tomiki (富木 謙治 Tomiki Kenji, 1900–1979) gave the name Shōdōkan (昭道館) to the dojo he founded in Osaka where he researched and promoted his unique method and style of teaching Aikido after resigning as technical director of the Aikikai. In deference to his wishes at the time not to have his name directly attached to his school, The term "Shodokan Aikido" was used in Japan to differentiate it from other schools of Aikido.
Shodokan (昭道館) is translated as 'the place for identifying the way'. It is made up of three kanji characters. Sho (昭) meaning 'bright' or 'clear', do (道) meaning 'road' or 'way', and kan (館) meaning 'building' or 'hall'. The first character, sho, also comes from the name of the company that provided Tomiki Kenji with the space for his dojo and is the first kanji of the Showa period in which the dojo was established (1967).
The logo used by the JAA (Japanese Aikido Association) is based on the symbol designed by Tomiki Kenji Shihan to represent the physiological and philosophical characteristics of Aikido. The symbol represents the functions of fire (red) and water (blue) combined with limitless space, represented by the white area. The following is a translation of his explanation of how he arrived at the design.
The original symbol form represents the dictum of the Kitō-ryū (起倒流?) school of Jujitsu 'the scroll of heaven', which maintains that the okiru (ki) or 'rise' is the manifestation of yo, and that taoru (to) or 'fall' is the manifestation of in. reference: 'Aikido & Randori', Scott Albright, 2002
ki (起) is the power of fire, represented by the red inner circle
tō (倒) is the power of water, represented by blue.
This eight directional star symbol can be seen widely incorporated in the logos, badges or kamon of many clubs and associations that teach and promote the Aikido of Kenji Tomiki Shihan. (some may be historical)
With the departure of Tetsuro Nariyama Shihan from the JAA (where he was technical director) and the subsequent formation of the SAF there is a bit of change in the use and association of the terms "Shōdōkan" and "Tomiki Aikido" as well as the use of the original symbol, nationally (in Japan) and Internationally (outside Japan).
Founded by Tomiki Kenji Shihan, the JAA have reverted to use of the original shodokan symbol without embellishment, but have moved to officially using the abreviation "Tomiki Aikido (富木合気道)" as their preffered style label in conjunction with the longer term "The Aikido of Kenji Tomiki (富木謙治の合気道)" to cement their link to the Tomiki Shihan legacy. It's relevant to note that the term "Tomiki Aikido" was almost exclusively used (in the west) by International associations, organisations and clubs to describe and identify their style. reference (http://aikido-kyokai.com/aikido/tomiki/)
Founded by Tetsuro Nariyama Shihan after resigning from the JAA, the SAF now exclusively uses the term "Shodokan (昭道館)" as their official style label for promoting the Aikido of Kenji Tomiki Shihan and for their honbu dojo. It's relevant and noteworthy that when Tomiki Kenji Shihan resigned from the Aikikai and formed the JAA, he founded a dojo in Osaka which he named as "Shōdōkan" (the place for identifying the way). Tetsuro Nariyama Shihan has had a great influence on the development of the Aikido of Kenji Tomiki during his long tenure within the JAA, and continues his work now as Chief Instructor of the Shodokan Dojo in Osaka and as President of the SAF whose distinctive logo/kamon notably has an International focus (Use of English Language).
Just as the original JAA symbol was and is still incorporated within international organisations logos, so too we are begininng to see the SAF symbol being adopted in the same manner.