Code for a physical binary clock with an Arduino, an LED dot matrix, and an RTC module.

binary_clock.ino

#include "LedControl.h"
#include <DS3231.h>

// Initialize the dot matrix and the clock
LedControl lc = LedControl(7, 5, 6, 1);
DS3231 clock;

void setup() {  
  // Start up and clear the matrix
  lc.shutdown(0, false);
  lc.setIntensity(0, 1);
  lc.clearDisplay(0);

  // Start the clock and set its time to the sketch compiling time
  clock.begin();
  clock.setDateTime(__DATE__, __TIME__);
}

void loop() {
  RTCDateTime dt = clock.getDateTime();
  
  lc.setColumn(0, 1, (dt.hour / 10) << 2);   // Tens digit of hour
  lc.setColumn(0, 2, (dt.hour % 10) << 2);   // Ones digit of hour
  lc.setColumn(0, 3, (dt.minute / 10) << 2); // Tens digit of minute
  lc.setColumn(0, 4, (dt.minute % 10) << 2); // Ones digit of minute
  lc.setColumn(0, 5, (dt.second / 10) << 2); // Tens digit of second
  lc.setColumn(0, 6, (dt.second % 10) << 2); // Ones digit of second

  delay(1000);
}