  Using world line fermions $\Upsilon_{\pm}^{m}=\Upsilon_{\pm}^{m}(\tau) $
valued in vector representation of $SO(d,4-d) $ with $d=2,3,4,$ we develop a
pure fermionic analog of Penrose twistor construction. First, we show that
Fermi antisymmetry requiring $(\Upsilon_{\pm}^{m}) ^{2}=0$ can be solved by
using twistor like variables. Then we study the corresponding dual twistor like
field action and show that quantum spectrum exhibits naturally 4D
$\mathcal{N}=1$ target space supersymmetry. Higher spin world line field
solutions of the constraint $(\Pi_{s}^{m}) ^{2}=0$, $s\in \mathbb{Z}$ are also
discussed.
