What is the use of interface constants?

The constant interface pattern is a poor use of interfaces. That a class uses some constants internally is an implementation detail. Implementing a constant interface causes this implementation detail to leak into the class's exported API. It is of no consequence to the users of a class that the class implements a constant interface. In fact, it may even confuse them. Worse, it represents a commitment: if in a future release the class is modified so that it no longer needs to use the constants, it still must implement the interface to ensure binary compatibility. If a nonfinal class implements a constant interface, all of its subclasses will have their namespaces polluted by the constants in the interface. 

There are several constant interfaces in the java platform libraries, such as java.io.ObjectStreamConstants. These interfaces should be regarded as anomalies and should not be emulated.

To avoid some pitfalls of the constant interface (because you can't prevent people from implementing it), a proper class with a private constructor should be preferred 

public final class Constants {

    private Constants() {
        // restrict instantiation
    }

    public static final double PI = 3.14159;
    public static final double PLANCK_CONSTANT = 6.62606896e-34;
}

And to access the constants without having to fully qualify them (i.e. without having to prefix them with the class name)

import static Constants.PLANCK_CONSTANT;
import static Constants.PI;

public class Calculations {

    public double getReducedPlanckConstant() {
        return PLANCK_CONSTANT / (2 * PI);
    }
}