If you cover and expose the form, the program draws the same Sierpinski gasket.

Private LastX As Single
Private LastY As Single

Private CornerX(2) As Single
Private CornerY(2) As Single

' Draw the Sierpinski gasket.
Private Sub Form1_Load(ByVal sender As System.Object, ByVal _
    e As System.EventArgs) Handles MyBase.Load
    Dim i As Integer

    Randomize()
    Me.Show()

    PickPoints()

    Dim gr As Graphics = Me.CreateGraphics()
    Do
        ' Pick the next corner.
        i = CInt(Int(Rnd() * 3))

        ' Move halfway from the current point
        ' to the new corner.
        LastX = (LastX + CornerX(i)) / 2
        LastY = (LastY + CornerY(i)) / 2

        gr.DrawLine(Pens.Red, LastX, LastY, LastX + 1, _
            LastY + 1)

        ' Check for events.
        Application.DoEvents()
    Loop
End Sub

Private Sub Form1_Closing(ByVal sender As Object, ByVal e _
    As System.ComponentModel.CancelEventArgs) Handles _
    MyBase.Closing
    End
End Sub

' Define the corner points.
Private Sub PickPoints()
    Const R As Integer = 5
    Dim i As Integer

    ' Clear.
    Dim gr As Graphics = Me.CreateGraphics()
    gr.Clear(Me.BackColor)

    ' Define the corners.
    CornerX(0) = R + 10
    CornerY(0) = Me.ClientRectangle.Bottom - R - 2
    CornerX(1) = Me.ClientRectangle.Width \ 2
    CornerY(1) = R + 2
    CornerX(2) = Me.ClientRectangle.Right - R - 2
    CornerY(2) = Me.ClientRectangle.Bottom - R - 2

    ' Draw the corners.
    For i = 0 To 2
        gr.DrawEllipse(Pens.Blue, _
            CornerX(i) - R \ 2, CornerY(i) - R \ 2, R, R)
    Next i

    ' Pick a starting point.
    i = CInt(Int(Rnd() * 3))
    LastX = CornerX(i)
    LastY = CornerY(i)
End Sub