Mars—Beyond the Red Horizon (31): Chapter 7 The Structure of Partial Isomorphism—Mars is Not Earth's Other (1)
In the preceding chapters, we have traced the history of Mars's observation, exploration, physical properties, evolutionary history, theories of life, and future plans. In this chapter, I would like to extract a single structure that runs through all of these and formulate it theoretically.
This book has called that structure "partial isomorphism."
But here, we must ask again. What is partial isomorphism? What does it express about Mars? And how does it connect to questions of music, literature, and imagination?
7.1 Between Isomorphism and Non-isomorphism—The Concept of Structural Correspondence
In mathematics, "isomorphism" means that a perfect correspondence exists between two structures. Two isomorphic structures are internally indistinguishable. What holds true for one also holds true for the other.
However, in the real world, perfect isomorphism hardly exists. It is natural that two planets, while within the same framework in terms of physical laws, may differ in their details.
The relationship between Mars and Earth is one of "not perfectly isomorphic, but within the same framework."
Mars is a terrestrial planet. It has a layered structure of core, mantle, and crust. There are traces of water. Carbon chemistry is established. There are seasonal changes. It uses sunlight as an energy source. In these respects, Mars is within the "isomorphic sphere" of Earth.
However, Mars lost its magnetic field. It lost its atmosphere. It lost its water cycle. It does not have plate tectonics. It could not form a biosphere. In these respects, Mars is "non-isomorphic" to Earth.
This book calls this relationship "partial isomorphism."
Partial isomorphism is a relationship where, while formed under the same physical laws, correspondence holds in one dimension but breaks down in another.
Mars is not perfectly isomorphic to Earth. But it is not completely unrelated to Earth either. It is precisely this "in-between" relationship that places Mars in a privileged position as an object of thought.
7.2 The Three-Layer Structure of Partial Isomorphism—What Corresponds and What Does Not
Mars's partial isomorphism can be considered by dividing it into three layers.
(A) Physical Isomorphism—Inside the Same Laws
This is the most fundamental layer. Mars and Earth exist under the same physical laws. Gravity, electromagnetic force, thermodynamics, chemical reactions. These all hold true in the same way on Earth and Mars.
The composition of the rocks is similar. Iron, silicon, oxygen, magnesium. These exist on both Earth and Mars. Water molecules (H₂O) exist (or existed) on both Earth and Mars.
In this layer, Mars is isomorphic to Earth.
(B) Temporal Non-isomorphism—Divergent Trajectories
The second layer is the difference on the time axis.
Mars and Earth were born at the same time. However, due to the difference in mass, Mars cooled down faster. It lost its magnetic field early, dissipated its atmosphere, and caused its water cycle to collapse.
In other words, Mars "began at the same time as Earth, but followed a different time axis."
This temporal divergence is not a spatial difference. It is not different because Mars is far from Earth, but because Mars has a different temporal structure than Earth.
Mars's present is neither Earth's past nor its future. Mars is a planet that exists in parallel with Earth while following a different time axis.
It is this temporal non-isomorphism that gives Mars a sense of "oldness." Feeling "traces of an ancient era" in the Martian landscape is not sentimentalism. It is a reflection of the scientific fact that Mars aged faster than Earth.
(C) Environmental Non-sustainability—The Collapse of Stability
The third layer is the problem of stability.
Earth has maintained environmental stability through complex feedback systems. Carbon cycle, water cycle, plate tectonics, magnetic field. These interact with each other to maintain Earth as a "planet where life can be sustained."
Mars could not possess this stability. The feedback systems did not function. Climate stability was lost. A biosphere was not formed, or even if it was, it was not maintained.
This "non-sustainability" is not merely a lack of function. It is a structure of having tried to sustain itself but failing to do so.
Mars is a planet that failed to sustain itself.
(Published on note on March 25, 2026)