A gyrocopter flying low over the sea at sunrise past a launch platform in The Last Caretaker, from the Update 4 press kit; it is not one of the Spires.
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Exploration

The Last Caretaker Spires: Spire of Command, Spire of Industry and Spire of Exodus

Short answer: The three Spires are tall rocks in the sea, each with a statue on top: the Spire of Command at 20, 46 (Augustus of Prima Porta), the Spire of Industry at 81, -94 (Vulcan, God of the Forge) and the Spire of Exodus at 183, -38 (Moses). Fly up with the Salvage Gyro or build Wall Steps, scan each statue with the Replica Scanner, and blow up the boulder below it to reach the cave. The three statues open The Fourth Chamber and its Oath Token.

Game version: Early Access, Update 6 (0.10.0)

Where the Spires Are

Where are the three Spires?

There are three, each a tall rock standing out of the sea with a statue on top: the Spire of Command at 20, 46, the Spire of Industry at 81, -94 and the Spire of Exodus at 183, -38.

No navigation beacon reveals the Spire of Exodus on the map, so sail to it by its coordinates.

See also: Spires on the interactive map

What are the Spires for?

Each statue is one of the three you need for The Fourth Chamber. Scan it to unlock it to build, then place all three inside the chamber to open it and reach the Oath Token.

See also: The Fourth Chamber and the Oath Token

What Is on Each Spire

What is on the Spire of Command?

The statue is Augustus of Prima Porta. The Spire also holds samples 598 Knowledge in Pieces, 604 The Social Wall and 641 The Shared Pursuit, the Compass and Meditation memories, two Decoy Deployment Pods, and plenty of Gold Ore with some Titanium Ore.

Watch your footing on the cliff sides; they are not safe.

Two small petrol canisters, 400 L in all, are also on it.

See also: what samples are for

What is on the Spire of Industry?

The statue is Vulcan, God of the Forge. The Spire also holds samples 603 Moving as One, 635 Feelings Made Visible and 647 Branching Thought, and five memories: Camera, Maps, Teddy Bear, Commander’s Log and Guitar. The cave behind the boulder has two Quantum Crystals and some cash.

Night Crawlers can come up in the water around it.

See also: what memories do for your humans

What is on the Spire of Exodus?

The statue is Moses. The Spire also holds samples 617 Thought Becomes Code, 624 Environmental Memory and 628 Between Human and Animal. A Talon Shark swims in the water around it.

Its rock is rich in ore: Gold, Graphite, Rutile, Hematite, Chromite, Cobalt and a few Quantum Crystal nodes.

See also: mining the ores

Getting Up and Into the Cave

How do you get to the top of a Spire?

Fly up with the Salvage Gyro, or build Wall Steps up the rock.

Landing the gyro on top is hard, the Spire of Exodus most of all: hover above the peak, cut the engine and jump out, and bring four or five petrol cans and repair materials for crashes.

See also: Salvage Gyro guide

How do you get into the Spire cave?

Just below each statue is a small cave whose mouth is blocked by a boulder. Blow the boulder up to get in; two Large Grenades do it at the Spire of Industry.

How do you scan the statue?

Scan the statue with the Replica Scanner, not the Photo Drone. The statue then shows up to build.

See also: how to use the Replica Scanner

Official sources and wiki