ESRI:54091

PROJCRS["WGS_1984_Peirce_quincuncial_North_Pole_diamond",
    BASEGEOGCRS["WGS 84",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["WGS_1984_Peirce_quincuncial_North_Pole_diamond",
        METHOD["Peirce Quincuncial (Diamond)"],
        PARAMETER["Latitude of natural origin",90,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",0,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",0,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",0,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Not known."],
        AREA["World."],
        BBOX[-90,-180,90,180]],
    ID["ESRI",54091]]

PROJCS["WGS_1984_Peirce_quincuncial_North_Pole_diamond",
    GEOGCS["GCS_WGS_1984",
        DATUM["D_WGS_1984",
            SPHEROID["WGS_1984",6378137.0,298.257223563]],
        PRIMEM["Greenwich",0.0],
        UNIT["Degree",0.0174532925199433]],
    PROJECTION["Peirce_Quincuncial"],
    PARAMETER["False_Easting",0.0],
    PARAMETER["False_Northing",0.0],
    PARAMETER["Central_Meridian",0.0],
    PARAMETER["Scale_Factor",1.0],
    PARAMETER["Latitude_Of_Origin",90.0],
    PARAMETER["Option",1.0],
    UNIT["Meter",1.0]]

proj4.defs("ESRI:54091","+proj=peirce_q +shape=diamond +lat_0=90 +lon_0=0 +k_0=1 +x_0=0 +y_0=0 +datum=WGS84 +units=m +no_defs +type=crs");

{
    "$schema": "https://proj.org/schemas/v0.7/projjson.schema.json",
    "type": "ProjectedCRS",
    "name": "WGS_1984_Peirce_quincuncial_North_Pole_diamond",
    "base_crs": {
        "name": "WGS 84",
        "datum": {
            "type": "GeodeticReferenceFrame",
            "name": "World Geodetic System 1984",
            "ellipsoid": {
                "name": "WGS 84",
                "semi_major_axis": 6378137,
                "inverse_flattening": 298.257223563
            }
        },
        "coordinate_system": {
            "subtype": "ellipsoidal",
            "axis": [
                {
                    "name": "Longitude",
                    "abbreviation": "lon",
                    "direction": "east",
                    "unit": {
                        "type": "AngularUnit",
                        "name": "Degree",
                        "conversion_factor": 0.0174532925199433
                    }
                },
                {
                    "name": "Latitude",
                    "abbreviation": "lat",
                    "direction": "north",
                    "unit": {
                        "type": "AngularUnit",
                        "name": "Degree",
                        "conversion_factor": 0.0174532925199433
                    }
                }
            ]
        }
    },
    "conversion": {
        "name": "WGS_1984_Peirce_quincuncial_North_Pole_diamond",
        "method": {
            "name": "Peirce Quincuncial (Diamond)"
        },
        "parameters": [
            {
                "name": "Latitude of natural origin",
                "value": 90,
                "unit": {
                    "type": "AngularUnit",
                    "name": "Degree",
                    "conversion_factor": 0.0174532925199433
                },
                "id": {
                    "authority": "EPSG",
                    "code": 8801
                }
            },
            {
                "name": "Longitude of natural origin",
                "value": 0,
                "unit": {
                    "type": "AngularUnit",
                    "name": "Degree",
                    "conversion_factor": 0.0174532925199433
                },
                "id": {
                    "authority": "EPSG",
                    "code": 8802
                }
            },
            {
                "name": "Scale factor at natural origin",
                "value": 1,
                "unit": "unity",
                "id": {
                    "authority": "EPSG",
                    "code": 8805
                }
            },
            {
                "name": "False easting",
                "value": 0,
                "unit": "metre",
                "id": {
                    "authority": "EPSG",
                    "code": 8806
                }
            },
            {
                "name": "False northing",
                "value": 0,
                "unit": "metre",
                "id": {
                    "authority": "EPSG",
                    "code": 8807
                }
            }
        ]
    },
    "coordinate_system": {
        "subtype": "Cartesian",
        "axis": [
            {
                "name": "Easting",
                "abbreviation": "E",
                "direction": "east",
                "unit": "metre"
            },
            {
                "name": "Northing",
                "abbreviation": "N",
                "direction": "north",
                "unit": "metre"
            }
        ]
    },
    "scope": "Not known.",
    "area": "World.",
    "bbox": {
        "south_latitude": -90,
        "west_longitude": -180,
        "north_latitude": 90,
        "east_longitude": 180
    },
    "id": {
        "authority": "ESRI",
        "code": 54091
    }
}

<?xml version="1.0" encoding="utf-8"?>
<Map srs="+proj=peirce_q +shape=diamond +lat_0=90 +lon_0=0 +k_0=1 +x_0=0 +y_0=0 +datum=WGS84 +units=m +no_defs +type=crs">
	<Layer srs="+proj=peirce_q +shape=diamond +lat_0=90 +lon_0=0 +k_0=1 +x_0=0 +y_0=0 +datum=WGS84 +units=m +no_defs +type=crs">
	</Layer>
</Map>