```python
import gi
gi.require_version('Gtk', '3.0')
from gi.repository import Gtk, Gdk
import hashlib

class PatternPasswordApp(Gtk.Window):
    def __init__(self):
        super().__init__(title="Pattern Password")

        # Window properties
        self.set_default_size(300, 300)
        self.set_border_width(10)

        # Create a drawing area for the pattern
        self.drawing_area = Gtk.DrawingArea()
        self.drawing_area.set_size_request(250, 250)
        self.drawing_area.connect("draw", self.on_draw)
        self.drawing_area.connect("button-press-event", self.on_press)
        self.drawing_area.connect("motion-notify-event", self.on_motion)
        self.drawing_area.connect("button-release-event", self.on_release)
        self.drawing_area.set_events(Gdk.EventMask.BUTTON_PRESS_MASK |
                                     Gdk.EventMask.POINTER_MOTION_MASK |
                                     Gdk.EventMask.BUTTON_RELEASE_MASK)

        # Grid settings
        self.grid_size = 3
        self.dots = [(x, y) for x in range(self.grid_size) for y in range(self.grid_size)]
        self.pattern = []
        
        # Set up the layout
        vbox = Gtk.Box(orientation=Gtk.Orientation.VERTICAL, spacing=6)
        vbox.pack_start(self.drawing_area, True, True, 0)
        self.add(vbox)
        
        self.show_all()

    def on_draw(self, widget, cr):
        # Draw the grid
        cr.set_line_width(2)
        cr.set_source_rgb(0, 0, 0)

        width = widget.get_allocated_width()
        height = widget.get_allocated_height()
        dot_size = min(width, height) // self.grid_size
        padding = (min(width, height) - dot_size * self.grid_size) // 2

        for x in range(self.grid_size):
            for y in range(self.grid_size):
                center_x = padding + x * dot_size + dot_size // 2
                center_y = padding + y * dot_size + dot_size // 2
                cr.arc(center_x, center_y, dot_size // 4, 0, 2 * 3.14)
                cr.fill()

        # Draw lines between the dots based on the pattern
        if len(self.pattern) > 1:
            cr.set_source_rgb(0, 0, 1)  # Blue lines
            cr.set_line_width(3)
            cr.move_to(padding + self.pattern[0][0] * dot_size + dot_size // 2,
                       padding + self.pattern[0][1] * dot_size + dot_size // 2)
            for dot in self.pattern[1:]:
                cr.line_to(padding + dot[0] * dot_size + dot_size // 2,
                           padding + dot[1] * dot_size + dot_size // 2)
            cr.stroke()

    def on_press(self, widget, event):
        # Start drawing the pattern
        x, y = event.x, event.y
        dot_size = widget.get_allocated_width() // self.grid_size
        self.current_dot = self.get_dot_at_position(x, y, dot_size, widget)
        if self.current_dot is not None:
            self.pattern.append(self.current_dot)
            self.draw_pattern(widget)

    def on_motion(self, widget, event):
        # Continue drawing while dragging the mouse
        if event.state & Gdk.ModifierType.BUTTON1_MASK:
            x, y = event.x, event.y
            dot_size = widget.get_allocated_width() // self.grid_size
            next_dot = self.get_dot_at_position(x, y, dot_size, widget)
            if next_dot is not None and next_dot not in self.pattern:
                self.pattern.append(next_dot)
                self.draw_pattern(widget)

    def on_release(self, widget, event):
        # End drawing, process the pattern and reset
        self.process_pattern()
        self.pattern.clear()  # Reset the pattern after releasing the mouse

    def get_dot_at_position(self, x, y, dot_size, widget):
        # Calculate which dot was pressed
        padding = (min(widget.get_allocated_width(), widget.get_allocated_height()) - dot_size * self.grid_size) // 2
        x_index = (x - padding) // dot_size
        y_index = (y - padding) // dot_size
        if 0 <= x_index < self.grid_size and 0 <= y_index < self.grid_size:
            return (x_index, y_index)
        return None

    def draw_pattern(self, widget):
        # Redraw the pattern
        widget.queue_draw()

    def process_pattern(self):
        # Hash the pattern to store or validate it
        pattern_str = ''.join(f"{x}{y}" for x, y in self.pattern)
        hashed_pattern = hashlib.sha256(pattern_str.encode()).hexdigest()
        print("Pattern Hash:", hashed_pattern)

if __name__ == "__main__":
    app = PatternPasswordApp()
    app.connect("destroy", Gtk.main_quit)
    Gtk.main()
```

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