A Self-Sustaining Gravity-Driven Hydraulic Oscillation System for Mechanical Wave Generation in Modular Game Environments

Technical Analysis and Proof of Concept

Jonathan Jones
Liana Banyan Corporation / Upekrithen LLC
January 2026

Abstract

This paper presents the theoretical foundation and engineering analysis of a novel self-sustaining hydraulic oscillation system designed for the HexIsle modular game table. The system employs three nested hexagonal reservoirs (X, Y, Z) operating under gravity-driven water transfer, regulated by a central water wheel escapement mechanism. Unlike traditional mechanical or electronic wave generators, this system achieves continuous alternating current (AC) hydraulic waves through purely gravitational and hydraulic means, with the water flow itself powering the timing mechanism. Mathematical analysis demonstrates that the system operates within safe engineering margins, with torque outputs exceeding requirements by a factor of 6×. The optimized design utilizes 420 hexagonal game tiles with a total system weight of approximately 320 pounds, making it practical for home assembly and use. The design draws upon established principles from communicating vessels, pendulum dynamics, and historical water clock escapement mechanisms, synthesizing them into a novel application for interactive gaming environments.

1. Introduction

1.1 Background and Motivation

The HexIsle game system represents a departure from conventional tabletop gaming by introducing dynamic environmental elements through hydraulic actuation. The fundamental challenge addressed in this paper is the creation of continuous, self-sustaining wave motion across a field of 420 modular hexagonal game tiles (“Hexels”) without reliance on external power sources or electronic control systems.

The design philosophy prioritizes mechanical elegance and self-regulation, drawing inspiration from historical water clocks (clepsydrae) that achieved remarkable precision through purely hydraulic means. As documented by historians of technology, Chinese engineers of the Song dynasty developed sophisticated water wheel escapement mechanisms as early as the 10th century, demonstrating that complex timing functions could be achieved through careful management of water flow and gravity.

The choice of 420 Hexels is deliberate: this number is divisible by 1, 2, 3, 4, 5, and 6, enabling flexible game configurations for any number of players from solo play to full six-player games with equal territory distribution.

1.2 System Overview

The HexIsle hydraulic system consists of four primary components operating in concert:

1. Reservoir X (Outer/Tabletop): A hexagonal water reservoir integrated into the game table surface, providing pressure head to the perimeter Hexels via six vertex columns.
2. Reservoirs Y and Z (Middle and Inner): Nested hexagonal reservoirs suspended beneath the tabletop that oscillate vertically, creating the AC wave pattern through water transfer.
3. Table Clock (Central Water Wheel): A water wheel escapement mechanism that regulates the oscillation frequency while being powered by the very water flow it controls.
4. Jug Port (Fill/Top-off System): A standard water cooler connection on the clock housing that accepts 5-gallon jugs for initial filling and automatic top-off.

The key innovation lies in the self-sustaining nature of the oscillation: water transferring between reservoirs Y and Z through the central clock mechanism creates an automatic seesaw effect, where the heavier reservoir drops, pushing water to the lighter one, which then becomes heavier and reverses the cycle.

2. Theoretical Foundation

2.1 Hydrostatic Pressure Principles

The system operates on the fundamental principle that hydrostatic pressure depends on fluid column height, not container geometry. This relationship is expressed by:

P = ρgh

Where P is pressure, ρ is fluid density (1000 kg/m³ for water), g is gravitational acceleration (9.81 m/s²), and h is the vertical height of the fluid column.

For the HexIsle system with a standard 3-foot table height as the effective head:

P = 1000 × 9.81 × 0.91 = 8,927 Pa ≈ 1.30 psi

2.2 Communicating Vessels and Oscillation

The principle of communicating vessels states that a homogeneous fluid in connected containers will settle to the same level, regardless of container geometry. This principle, known since antiquity and formalized through Pascal’s law, forms the basis for the Y-Z reservoir interaction.

When two connected vessels contain different fluid volumes, pressure differential causes flow until equilibrium. However, if the vessels are mechanically constrained to move vertically, and water transfer is regulated by a flow restriction (the clock mechanism), the system can achieve stable oscillation rather than immediate equilibrium.

The oscillation period T for a U-tube water column is given by:

T = 2π√(L/2g)

Where L is the total length of the water column. This relationship demonstrates that oscillation frequency is independent of cross-sectional area and depends only on column length and gravity.

2.3 Escapement Mechanism Theory

The water wheel escapement in the HexIsle system serves the same fundamental purpose as escapements in mechanical clocks: converting continuous energy flow into discrete, regulated increments. Historical precedent exists in the astronomical clock towers of Song dynasty China, where water wheels with steelyard clepsydra devices achieved precise timekeeping through water flow regulation.

The key insight is that the escapement mechanism need not be externally powered; the water flowing through the system to transfer between Y and Z reservoirs provides the energy to drive the water wheel, which in turn regulates that very flow. This creates a closed-loop self-regulating system.

3. System Architecture

3.1 Nested Reservoir Configuration

The three-reservoir system (X, Y, Z) is arranged concentrically beneath the hexagonal game table. The tabletop reservoir (X) spans the full 60-inch diameter field and maintains connection to the Hexel array through the perimeter distribution system.

The critical weight relationship Y + Z > X ensures that the combined inner reservoirs can counterbalance the outer reservoir, enabling the self-sustaining oscillation cycle. With the table structure and Hexels adding approximately 150 lbs, total system weight is approximately 320 lbs.

3.2 Water Flow Pathways

Water flows through the system via three distinct pathways:

• Perimeter Path: From reservoir X through six vertex columns to horizontal perimeter pipes, feeding edge Hexels through ball valve connections.
• Center Path: From reservoirs Y and Z through the Table Clock mechanism to center Hexels.
• Inter-Hexel Path: Between adjacent Hexels via Swan Neck connectors, propagating the AC wave across the field.

3.3 Table Clock Mechanism

The Table Clock serves three simultaneous functions:

1. Flow Regulation: The water wheel provides variable resistance to water flow, controlling oscillation frequency.
2. Energy Harvesting: Water flowing between Y and Z drives the wheel, powering the clock display.
3. Time Display: The wheel axle drives a visible clock face, providing game timing functionality.

This design embodies the principle “when you stop the clock, you literally stop the ocean” - blocking flow through the clock mechanism halts the Y-Z oscillation and freezes all wave activity.

3.4 Jug Port Fill System

A standard 55mm water cooler fitting on the center top of the clock housing accepts inverted 5-gallon water jugs. This serves multiple functions:

• Initial Fill: Water flows by gravity until system is full.
• Automatic Top-off: Spring-loaded valve opens under vacuum when system loses water to evaporation.
• Pressure Boost: Jug at 24" above table adds 0.87 psi to operating pressure.
• Visual Indicator: Jug level provides visual feedback on system health.

4. Hexel Actuation Mechanism

4.1 Hexel Count Optimization

The system utilizes 420 Hexels, a number chosen for its divisibility properties. Being divisible by 1, 2, 3, 4, 5, and 6 (LCM = 60, and 420 = 7 × 60), it enables equal territory distribution for any number of players:

4.2 Golden Lotus Flow Converter (Design #3)

Each Hexel contains a Golden Lotus mechanism that converts bidirectional hydraulic AC flow into unidirectional rotation. The Design #3 configuration places Tesla valve cups in the center pointing outward, with a 36-vane rotor ring on the outer perimeter.

4.3 Unidirectional Rotation Principle

The Tesla valve cups are angled 30° clockwise from radial at their exit points. This geometry ensures that both push and pull phases of the AC wave create rotation in the same direction:

• Push Phase: Cups 1, 3, 5 (UP orientation) expel water tangentially, striking rotor vanes and driving clockwise rotation.
• Pull Phase: Cups 2, 4, 6 (DOWN orientation) expel water tangentially, also driving clockwise rotation.

The Rooster Teeth inside each cup (1.5mm protrusions, 6mm tall) catch the flow and amplify torque during push strokes while Tesla valve curves minimize reverse-flow efficiency during pull strokes.

5. Torque Analysis and Verification

5.1 Force Calculation

The total active vane area in the rotor ring:

A = 36 vanes × (5mm × 12mm) = 2,160 mm² = 3.35 in²

Force generated at 1.30 psi operating pressure:

F = P × A = 1.30 psi × 3.35 in² = 4.36 lbs

5.2 Torque Generation

With average moment arm of 25mm (0.98 inches):

τ = F × r = 4.36 lbs × 0.98" = 4.3 in-lb (gross)

Accounting for 70% efficiency (Tesla valve losses, turbulence):

τ_net ≈ 3.0 in-lb

5.3 Load Requirements

The Hexel mechanism must overcome friction in the gear train (Ouralis → Pgears → Main Gear) and drive the Cradle wave mechanism plus floating game pieces. Estimated total load:

τ_required ≈ 0.5 in-lb

5.4 Safety Margin

Safety Factor = τ_net / τ_required = 3.0 / 0.5 = 6×

This 6× safety margin provides adequate overhead for manufacturing variations, debris accumulation, and heavier-than-expected game pieces, while keeping the system weight practical for home use.

6. System-Wide Hydraulic Validation

6.1 Volume Analysis

Total stroke volume for all 420 Hexels:

V_stroke = 420 Hexels × 5 mL/Hexel ≈ 2.1 liters = 0.55 gallons

With 20 gallons total reservoir capacity, pressure remains stable throughout operation with minimal level fluctuation (ratio of approximately 36:1).

6.2 Pressure Distribution

The parallel flow architecture ensures uniform pressure across all Hexels. Friction loss calculation for the longest path (center to edge, 30 inches):

Pressure at furthest Hexel: 1.30 - 0.06 = 1.24 psi (95% retention)

6.3 Summary of Validation Results

7. Historical Precedent and Prior Art

7.1 Water Clock Technology

The HexIsle Table Clock mechanism draws upon millennia of water clock development. The earliest clepsydrae date to c. 1500 BCE in Egypt and Babylon. By the 3rd century BCE, Greek engineers including Philo of Byzantium had developed escapement mechanisms for water clocks, while Ctesibius incorporated gears and feedback systems.

Chinese innovations proved particularly relevant: Zhang Heng (78-139 CE) introduced compensating tanks for pressure regulation, while Yi Xing and Liang Lingzan (725 CE) created the first water wheel linkwork escapement. Su Song’s monumental astronomical clock tower (1088 CE) demonstrated that complex mechanical systems could be powered and regulated entirely by water flow.

7.2 Modern Hydraulic Oscillators

Contemporary research in microfluidic hydraulic oscillators (MHOs) confirms the viability of self-switching fluidic circuits. Studies published in Nature Communications demonstrate gravity-driven oscillators achieving stable periods from 0.4 seconds to 2 hours using only water head pressure. These devices convert constant pressure inputs to pulsatile outputs through passive fluidic elements, validating the fundamental principle underlying the HexIsle system.

7.3 Novel Contributions

While individual elements of the HexIsle system have historical precedent, the synthesis represents a novel application:

• Integration of three nested oscillating reservoirs for AC wave generation
• Water wheel escapement driven by the flow it regulates (closed-loop self-regulation)
• Tesla valve-based unidirectional converters in each actuator module
• Standard water cooler jug integration for fill and maintenance
• Application to modular gaming environment with 420 parallel actuators

8. Manufacturing and Distribution

8.1 Distributed Production Model

The HexIsle system is designed for distributed manufacturing through a network of local production nodes. Components are produced using lithographic processes that enable high precision at low cost, with designs made available to community members for personal production.

The production chain consists of three tiers: design origination (Liana Banyan Corporation), component production (distributed Nodes), and final assembly (local Assemblers). This model aligns with cooperative economics principles, ensuring fair compensation at each stage while minimizing shipping distances and costs.

8.2 Flat-Pack Shipping Configuration

The assembled table ships in a collapsed configuration secured by a single zip-tie. Telescoping legs fold flat, and the reservoir structure nests within the table frame. Upon receipt, customers simply:

1. Snip the zip-tie to release the collapsed configuration
2. Pull upward to extend the telescoping legs
3. Level the table using adjustable feet
4. Invert a standard 5-gallon water jug onto the jug port
5. Open bleed valves to release trapped air during filling
6. Close bleed valves when system is full; play begins automatically

9. Conclusions

This analysis demonstrates the theoretical viability of the HexIsle gravity-driven hydraulic oscillation system. The three-reservoir configuration (X, Y, Z) with central water wheel escapement achieves self-sustaining AC wave generation through purely mechanical and hydraulic means.

Key findings include:

• Operating pressure of 1.30 psi (3-foot table height) provides 6× torque safety margin
• 420 Hexels enable flexible player configurations (divisible by 1-6)
• Total system weight of approximately 320 lbs is practical for home use
• Parallel flow architecture maintains 95% pressure retention across all Hexels
• Design #3 (outer rotor, inner Tesla valves) achieves ~3.0 in-lb net torque per Hexel
• Self-sustaining oscillation requires only initial filling; no external power input
• Standard 5-gallon water jug provides fill, top-off, and backup functionality
• Clock mechanism provides both timing regulation and game time display

The design synthesizes established principles from hydrostatics, pendulum dynamics, and historical water clock escapements into a novel application that enables dynamic, interactive gaming environments without electronic components. The distributed manufacturing model ensures accessibility while maintaining quality and fair economic distribution.

References

1. Kim, S.J. et al. “Multiple independent autonomous hydraulic oscillators driven by a common gravity head.” Nature Communications 6, 7301 (2015).
2. Needham, J. Science and Civilisation in China, Vol. 4: Physics and Physical Technology. Cambridge University Press (1965).
3. Price, D.J. de Solla. “On the Origin of Clockwork, Perpetual Motion Devices and the Compass.” United States National Museum Bulletin 218 (1959).
4. Lin, T.Y. “Historical development of water-powered mechanical clocks.” Mechanical Sciences 12, 203-217 (2021).
5. White, F.M. Fluid Mechanics, 8th Edition. McGraw-Hill Education (2016).
6. Wikipedia contributors. “Water clock,” “Escapement,” “Communicating vessels.” Wikipedia, The Free Encyclopedia (accessed January 2026).

Patent and Intellectual Property Notice

The HexIsle hydraulic game table system and associated mechanisms described in this paper are the subject of pending provisional patent applications filed by Liana Banyan Corporation and Upekrithen LLC. Application numbers include 63/925,672 and 63/938,216.

This document is published for academic and technical review purposes. Commercial reproduction or implementation of the described systems requires appropriate licensing arrangements.

© 2026 Liana Banyan Corporation / Upekrithen LLC
All Rights Reserved