Cut cake into four 1-inch layers. Whip sweetened whipping cream until stiff. Put layers together with each of the three different fillings.
Bottom Layer: Pineapple Filling: Soften 2 teaspoons gelatin in 1 tablespoon pineapple syrup. Dissolve over hot water. Stir into crushed pineapple. Tint with green food coloring. Chill. When partially set, fold in 1 cup sweetened whipped cream.
Middle Layer: Apricot Filling: Soften 2 teaspoons gelatin in 1 tablespoon apricot syrup. Dissolve over hot water. Stir into mashed apricot pulp. Chill. When partially set, fold in 1 cup sweetened whipped cream.
Top Layer: Raspberry Filling: Soften 2 teaspoons gelatin in 1 tablespoon fruit juice. Dissolve over hot water. Stir into raspberry jam. Chill. When partially set, fold in 1 cup sweetened whipped cream.
Cover top and sides of cake with remaining whipped cream. Sprinkle with pistachio nuts. Chill for 3 hours before serving. Arrange apricot halves, rounded sides out, around base of the cake. Add a few green leaves. Serve.