On Spherical t-designs in ℝ2

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Let X be a spherical t-design in ℝ2 with |X| = n. It is shown that:

  • (i)

    for t + 1 ⩽ n ⩽ 2t + 1, X must be a regular n-gon

  • (ii)

    for n = 2t + 2, X is the union of two regular (t+1)-gons

  • (iii)

    for each n⩾2t+3, there are many (ℵ1) X's which can not be decomposed into the union of regular ki-gons where ki ⩾ (t + 1).

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