[1][2] He is most well known for his book Ge xiang xin shu (革象新书), translated either as New Elucidation of the Heavenly Bodies or New Writing on the Image of Alteration, wherein he described a new method to calculate pi.
[2] The Shangyangzi jindan dayao says that, as a child, he was injured during the war between Mongol leader Kublai Khan and the Song dynasty.
[2][1] Zhixu's biography says that Zhao was a Daoist hermit, and later a patriarch of the Quanzhen (Complete Perfection) School of Song-Yuan Daoism.
[1] The biography further claims that Zhao gave a manuscript of Ge xiang xin shu to his disciple, Zhu Hui, while on his deathbed on Jiming mountain in the Zhejiang Province.
[4][5] He also described an instrument used to calculate the angle between a star and the North Pole, and another used to find the differences in the right ascension of two celestial objects.
[2] In Ge xiang xin shu (革象新书), Zhao described an optical experiment he undertook in a chapter titled "Pinhole Image".
"[2][3] He also wrote at least two Daoist books, Xian Fo tong yuan [lun] ([A Discourse] on the Common Origins of [the Teachings of] Immortals and Buddhas) and Jin dan nan wen (Difficult Problems of Gold Cinnabar).