Significance of fasting on the feast of the Elevation of the Holy Cross

There are four fasting periods in the Church year, which lead to the feasts of Apostles Peter and Paul, the Dormition of the Mother of God, the Nativity of the Lord, and, of course, Great Lent and Holy Week which lead to Pascha. The eve of the beheading of St John the Baptist is a strict fast day in many Orthodox traditions as well.